Qur'an did not, does not, and will not rely on manuscripts. It relies on memorization:
بَلْ هُوَ آيَاتٌ بَيِّنَاتٌ فِي صُدُورِ الَّذِينَ أُوتُوا الْعِلْمَ ۚ وَمَا يَجْحَدُ بِآيَاتِنَا إِلَّا الظَّالِمُونَ
Rather, the Qur'an is distinct verses [preserved] within the breasts of those who have been given knowledge. And none reject Our verses except the wrongdoers.
— Surat Al-'Ankabut 29:49
This is a matter that has had no disagreement among scholars of Islam at any time. Until today, reciters are required to get ijazah, which is a license to recite Qur'an, that involves reciting the Qur'an word for word and letter for letter in front of a teacher, and the document of the ijazah would have the complete chain of teachers from that person till the Prophet ﷺ. In addition, in Islam, Qur'an is preserved the other way round: It is memorized, and what is memorized is captured through writing, then the writing is validated through memorizers (rather than memory being validated through manuscripts), as per the following source, for example:
مذهب الأصوليين وفقهاء المذاهب الأربعة والمحدثين والقراء: أن التواتر شرط في صحة القراءة، ولا تثبت بالسند الصحيح غير المتواتر، ولو وافقت رسم المصاحف العثمانية، وهو قول محدث لا يعول عليه، ويؤدي إلى تسوية غير القرآن بالقرآن
— NOTE: My own translation, so treat with care:
The school of the scholars of the principles [of jurisprudence] and the jurists of the four schools of jurisprudence and the scholars of hadith and the scholars of recitation that tawātur (successive narration) is a condition for a correct recitation; what is not mutawatir even when authentic is not proof, even if a recitation [that is not mutawatir] matches a manuscript of the 'Uthmanic mus'hafs, as this would be an innovation that cannot be taken into account, as it would be equating non-Qur'an with Qur'an.
— Muqaddimāt fi 'Ilm al-Qirā'at, pp. 71
Assuming that the Birmingham manuscript contains differences to what is memorized, then it means the manuscript is wrong. Whether it is an exact replica of today's Qur'an or completely different neither affirms nor disproves the accuracy of the Qur'an.