My answer is primarily based on this nice history of spreading of the qiraa't of the scholar ibn Amin. Which shows that the qiraa'a of Hafs 'an 'Asim has not even well known or prevalent in al-Kufa itself for many centuries. I tried to summarize it as much as i can.
Note that among the qurraa' of the 7-10 mutawatir and 4 non-mutawatir readings which have somehow survived generally both the teachers and their students are also accepted and trustworthy hadith narrators, with exceptions such as Hafs who have been accused to be da'if, rejected, a fabricator etc.
Al-Bukahri has even reported that abu Bakr ibn 'Ayash أبو بكر بن عياش (Sho'aba) who should be his mate as students of 'Asim even denied to even know him! (See here in al-'Oqayly's ad-Do'afa')
The spread of the qiraat according the "old geographical" regions of the Islamic world
The Caliph 'Omar ibn al Khattab has sent some sahaba (May Allah be pleased with all of them) to the different regions like abu Musa al-Ash'ary to Basra and Abdullah ibn Mas'ud (al hudali) to Kufa, abu ad-Darda' to a-Shaam ... to teach the Qur'an!
a-Shaam (The region around Syria)
Until the Caliph 'Othman send the mushaf al-Imam or the mushaf copy of a-Shaam with al-Mughira ibn abi Shihab المغيرة بن أبي شهاب the people where reciting according the qiraa'a of abu ad-Darda'. But there have been some issue's where they still kept the choice of abu ad-Darda'. Afterwards when they adapted the qiraa'a of 'Othman (according the mushaf) so the qiraa'a of ibn 'Amir ابن عامر (one of the 7 quraa') has been influenced basically by 'Othman's and in cases by abu ad-Darda's and Nafi's نافع (an other one of the 7 quraa') qiraa'a. According to imam ibn al-Jazari (our major source when it comes to the 10 qiraa't nowadays) this qiraa'a was the major and one could say the official qiraa'a in the region until around 500 a.H. when an excellent qari' of ad-Duri 'an abi 'Amr الدوري عن أبي عمرو came from 'Iraq and introduced this qiraa'a which has afterwards widely spread in the country until the 'Othmans came and "imposed" the qiraa'a of Hafs 'an 'Asim.
Because of the proximity to al-Hijaz the qiraa'a of the people of Basra have been influenced by the qiraa'a of al-Hijaz so at the end of the 2nd hijri century people there where either following the qiraa'a of abu 'Amr أبو عمرو (which was influenced by the hijazi qiraa'at) or Ya'qub يعقوب (a pure basri qiraa'a). This last qiraa'a became more and more influence until it was the major one at the 5th century (this confirms a quote of ad-Dani from the Arabic wikipedia). But afterwards the qiraa'a of abu 'Amr again regained influence in Basra and the most parts of the Islamic world until the Turks "made an end" to its influence.
The influence of ibn Masu'ds qira'a
Ibn Mujahid quoted a statement of Sulayman al-A'mash (died 148 a.H.) trying to explain the influence of ibn Masu'ds (note he was hudali not qurashi) qiraa'a even after the mushaf al Imam (here referred to as qira-at Zayd ibn Thabit) has been sent to them -my own translation please take it carefully-:
I saw (experienced) the people of al-Kufa and the qiraa'a of Zayd (ibn Thabit) among them is not much more then the qiraa'a of Abdullah (ibn Masu'd) among you, only one or two (to express a few) men are reading it.
أدركت أهل الكوفة وما قراءة زيد فيهم إلا كقراءة عبد الله فيكم، ما يقرأ بها إلا الرجل والرجلان.
Note that the qiraa'a of ibn Masu'd (since permalinks seemingly can't be set for paragraphs of wikipedia articles, please check if the article shared later ist still existing in the actual version else jump back here) was not only in the harf/language of hudail but also containing some abrogated ahruf etc., and he at first rejected the mushaf al-Imam but accepted it later when he returned to Medina (Some scholars say that the person who asked 'Aisha in this hadith was following the qira'a of ibn Masu'd). 'Othman has sent Abdarrahman as-Sulami عبد الرحمن السلمي, (whom has been a student of Ali ibn abi Talib the 4th Caliph) with the mushaf copy to al-Kufa. Note that the Kufis for political reasons (their hate towards al-Hajjaj ibn Yousuf a-Thaqafi) rejected the qiraa'a of 'Asim (who followed the qurashi mushaf and the orders of al-Hajjaj) and kept the qiraa'a of ibn Masu'd.
The qari's of al-Kufa
So 'Asim ibn Bahdala (also known as 'Asim ibn abi an-Nujud or 'Asim al-Qaari' عاصم بن أبي النُّجُود أو عاصم القارئ ) chose a middle way between both the qiraa'a of hijaz and the qiraa'a of ibn Masu'd, but the influence of ibn Mas'ud's hudali dialect is still clear in this qiraa'a -in total opposite to the people of Hijaz- it pronounces the hamza widely (for example it's more likely for hijazi to say an-Nisa النسا -which is considered as more fasih- instead of an-Nisa' النساء, but a clearer example for this is ata'morona أتأمرون (do you order) which is also read atamorona أتامرون or a-dhi'b الذئب(the wolf) which is also read a-dheeb الذيب).
Later Hamzah (a persian qaari' and teacher of the other Kufi qaari' Khallaf) made a mix of all the Kufi qiraa'at, so his qiraa'a is influenced by 'Ali, ibn Masu'd and in cases Ubay's qiraa'a and it goes ahead with the mushaf of 'Othman.
Ibn Mujahid pointed out that the qiraa'a of 'Asim was spread in Kufa even if it was not the major qiraa'a, and people who followed it later preferred the riwaya of abu Bakr ibn 'Ayach (Sho'aba) أبو بكر بن عيَّاش as he has been approved with the qiraa'a many times by 'Asim, but abu Bakr was a withdrawn man, so that his riwaya became rare in Kufa and the riwaya of Hamzah grow instead and this was the major qiraa'a in Kufa at his (ibn Mujahid's) time. And also as it was the case for a-Shaam the qiraa'a of abu 'Amr came from Basra and was adapted there for many centuries until the 'Othmans came and "imposed" Hafs 'an 'Asim. And from the south of 'Iraq (especially) Karbala' the qiraa'a of 'Asim spread to the regions with a Shi'a majority.
Northern Africa and al-Andalus
The first reading which was widely spread there was the qiraa'a of ibn 'Amir by some preachers like Isma'il ibn Abdullah ibn abi al-Muhajir إسماعيل بن عبد الله بن أبي المهاجر. Note that most of the soldiers or fighters who have been part of the conquest of al-Andalus came from a-Shaam and brought the madhab of Imam al-Awza'ii there so the people of Maghrib and al-Andalus have known the (shaami) riwaya of Hisham 'an ibn 'Amir هشام عن ابن عامر first. And in the 2nd hijri century the qiraa'a of Hamzah reached al-Qayrawan and only a few people used to read according the qiraa'a of Nafi' according to Hind Shalabi in her book al-Qiraat fi Ifriqya this is due to the fact that the scholars or intellectual people followed imam Sufyan at-Thawri -who himself was a student of Hamza- and even if later the Aghalibah Dynasty (184 aH-296 aH) adopted the madhhab of abu Hanifa as an official madhhab the qira'a of 'Asim (who was among the teachers of abu Hanifa) was only one of many other qira'at there.
Ad-Dani (an other major source for the qiraa't and qur'an sciences, no continuous sanad of any qiraa'a which exists nowadays wouldn't have him in the narrator chain) quoted that ibn Barghout (272 a.H.) was the first one to impose the qiraa'a of Nafi' in al-Qayrawan. Hind Shalabi in her book (p. 232) corrected this information saying that qari' ibn Barghut was ordered by the Qadi (Maliki certainly) 'Abdullah ibn Talib عبد الله بن طالب not to teach any other harf (letter) than that of Nafi'i to his students. The people of Tunis chose the riwaya of Qalun while the western parts of al-Maghrib have chosen the riwaya of Warsh min tariq al-Azraq (a student of Warsh from Medina) رواية ورش المصري، من طريق يوسف الأزرق المدني. So one could say that this qiraa'a came more or less with the madhab of Imam Malik from Medina to the Maghrib and Andalus.
When did the riwaya of Hafs began to spread?
The end of the time when Islamic science and scholars were still creative
The historical reports above show that even if Hafs 'an 'Asim was one of the riwayat of the 7 qiraa't which were "defined" by ibn Mujahid, it was not prevalent in any region, but clearly well known by quraa' of the 7 or 10 qiraa't.
To find out when, how and from where it began to spread, we need more facts, in this could only be found based on reports of scholars or in earlier copies of tafsir books, because tafsir books usually had one of the regionally most prevalent qiraa'a as a basis... Yes now almost all tafsir books are based (or have it in the margin) on Hafs 'an 'Asim, but this was not the case only a few hundred years ago!
Ibn al-Jazari (751-833 a.H.) who lived in the days of Timur said that the official reading of a-Sham, al-Hijaz, al-Yaman and Egypt was ad-Dury 'an abi 'Amr. He didn't quote Khorasan or Turkey even if he studied there a long time. But Imam az-Zamakhshari (from Khorasan, died 538 a.H.) wrote his tafsir based on the reading of abu 'Amr. The same was the case with tafsir al-Wahidi (author of three tafsirs and the famous "asbab an-Nuzul" أسباب النزول died 468 a.H.). While abu As-Su'ud (died 952 a.H. -from Turkey) used Hafs as a basis for his tafsir. And outside of Turkey?
Imam a-Shawkani الشوكاني (from Yemen, died 1250 a.H.) used riwayat Qalun 'an Nafi' for his tafsir (called fath al-Qadir فتح القدير) and his sheikh Assan'ani الصنعاني did the same before, while Hadramaut was following the riwaya of ad-Duri.
The period of ignorance and stagnation and the copiers and commentators
A first sign for a tafsir based on Hafs in Turkey is the tafsir of Badr a-Deen al-Monshi ar-Rumi al-Hanafi 981 a.H. محمد بن بدر الدين المُنْشِي الرومي الحنفي. Later a scholar Mohammad al-Mar'ashi 1150 a.H. states that the riwaya of Hafs was in his country (south of Turkey) the standard. So one could say that after the spreading of riwayat hafs in the 16th century in Turkey it began to spread in a-Sham in the 18th and Egypt in the 19th century. The great hanafi scholar ibn 'Abideen a-Dimashqi ابن عابدين (1198 -1252 a.H./1748-1836) quotes that hanafi scholars have chosen the riwaya of Hafs 'an 'Asim and ad-Duri 'An abi 'Amr.
"ومشايخنا اختاروا قراءةَ أبي عَمْرٍو وحفصٍ عن عاصم"
Still at the time sheikh al quraa' of Dimashq/Damascus used to read the qiraa'a of abu 'Amr, and in 1951 still this was the major reading in Hadramaut.
- One major reason -according my source- for this (spread of Hafs) is that after ibn al-Jazari (a-Dimashqi) there haven't been any scholars who did some efforts in writing books on the issue of qiraat, so there have been a kind of ignorance for this science in a-Sham especially and in almost all the Islamic world, as the next great scholar on the matter from a-Sham was Ahmad al-Hilwani al-Kabir (1228-1307 a.H.) who, when visiting Mekka the sheikh of qiraat Ahamd al-Marzoqi recited for him in Hafs, which may indicate that this qiraa'a was the preferred qiraa'a at the time.
The well known reciter Abdarrashid as-Sofi from Somalia said that the qiraa'a of Hafs has only began to spread in the world in the 2 last centuries. As the people of al-Mashriq from Egypt, a-Shaam, al-'iraq, al-Hijaz, the Arabic Peninsula, Yemen and Sudan only knew the qiraa'a of abu Amr al-Basri in both riwayatt of ad-Duri and as-Susi. The major evidence is that most tafsir books had this qira'a like tafsir al-Jalalyn ... (I myself know a Somali student of him whom learnt both the riwaya of as-Susi 'an abi 'Amr and Hafs from him)
- An other reason is that when the printing plants at the time of 'Othmans started to come in use they have chosen the easiest riwaya for the basic or archaic machines to print mushafs. And the choice came to Hafs 'an 'Asim. But people didn't leave their qiraa'a so some even returned their mushafs because they were thinking that they have errors!
Later many countries followed the example and kept the riwaya of Hafs as an official reading.
The long article also includes an anecdote of Algerian salafis who stole/toke the masahif of warsh from mosques (in their country) and replaced it by the masahif Hafs which are distributed for free by Saudi-Arabian authorities.
The riwaya of Hafs has started spreading to become the official riwaya of the Turkish 'Othmans in the 16th century and, because of the archaic printing possibilities been chosen for a first Quran print, that's a major -technical- reason why this riwaya is now so prevalent in the Islamic world.
The other major reason is that after ibn al-Jazari no scholar -in the middle east- has made efforts to improve or publish in the science of qiraa'at for a long period of time, so that only a few people kept in touch with the other qiraa'at and riwayat which were not common in the regions of the Islamic world.
"I've been told by a Salafi-Brother that a well known Saudi scholar when praying behind a scholar from Mauritania began to correct the recitation of the imam, as he didn't know about any other riwaya except Hafs and thought Warsh was not correct!" I don't know how to qualify this statement! But maybe it shows the impact of the absence of scholars of qira'aat in the middle east!
Sheikh ibn Amin also wrote an article which is somewhat against the rawy Hafs where he gathered a lot of narrations and statements showing the inferiority of this rawy (whom he even try to show off as a weak narrator and fabricator for example he added an exclamation of Sho'aba asking where and when this Hafs might have learnt the reading of 'Asim, as he never saw nor met him with 'Asim). All this might be explained by mistakes for example Hafs was living in the house of 'Asim as his mother was 'Asims wife, the name Hafs ibn Sulayman al-Kufi is common so some Hadith scholars might have had confused him with another narrator.
Ibn Amin also said that it is a shame that nowadays riwayat Hafs appears in the margin of tafsir a-Tabari who in many cases rejected this riwaya or considered it as weak or less expressive. For example -I found it by coincidence, so there might be much better and clearer statements- in his interpretation of verse (2:165) he clearly holds a position against the qira'a (by Kufi, Makki and Basri scholars):
لو يرى (if they see/consider)
and prefers the qira'a (by Shami and Madani scholars):
ولو ترى (if you see)
He added that neither al-Qurtobi nor at-Tabari (who was a teacher of ibn Mujahid and from Kufa and therefore should exactly know the qira'at in there) who were well-versed in the science of qira'at addressed the special cases of recitation in riwayat Hafs in their tafsir and counted 24 singularity in his riwaya were no accepted riwaya has a back up for.
There are also accounts saying that the orientalists had some influence.
See also this Article in Arabic on the Issue.
I'd like to add that I also came across a Shi'a article on the Issue, but they only quoted sunni sources (like ibn Mujahid) until the 3rd hijri century. Some scholars say that Shi'a have only paid attention to this science after that time.
And Allah knows best!