All perfect praise be to Allah
Day of judgement will come when there will not be even single believer on the earth!! If you read anything(from quran and hadees) regarding when the end day will come, you will find that all of that signs indicate that islam will be less followed, even one of the last signs include the quran will be lifted by Allah, all the written copies will be vanished, and all people who remember anything from it will forget it. And this will be by Allah.
So, clearly its not the case that all people will be muslim, infact it will be reverse of it.
Al-Daarimi narrated saheeh isnaad (no. 3207) that ‘Abd-Allaah ibn
Mas’ood said: “Recite the Qur’aan much before it is taken away.” They
said, “These Mus-hafs will be taken away! What about that which is
(memorized) in men’s hearts?” He said: “Something will come and take
it one night, and in the morning they will wake up without it. They
will forget the phrase ‘laa ilaaha ill-Allaah’ and they will start to
recite the sayings and poetry of the Jaahiliyyah. That is when the
Word will be fulfilled against them.”
I think they might be pointing to a belief in islam that before judgement day, islam will become only religion in the world for some time. And islam will rule the whole world. And most of the scholars say it will be after Allah will send the Essa/jesus in the time of mehdi and they will kill dajjal and all that. But that will be before judgement day, not JUST BEFORE... like many other signs which have already been fulfilled they were supposed to happen before judgement day, not JUST BEFORE judgement day. You might talk with some local scholar to clarify about this belief in your area. whether it is about just before judgement day or this happening mentioned in ahadees.
Narrated AbuHurayrah: The Prophet (peace_be_upon_him) said: There is
no prophet between me and him, that is, Jesus (peace_be_upon_him). He
will descent (to the earth). When you see him, recognise him: a man of
medium height, reddish fair, wearing two light yellow garments,
looking as if drops were falling down from his head though it will not
be wet. He will fight the people for the cause of Islam. He will break
the cross, kill swine, and abolish jizyah. Allah will perish all
religions except Islam. He will destroy the Antichrist and will live
on the earth for forty years and then he will die. The Muslims will
pray over him.
(Book 37, Number 4310: Abu Dawud)
Read some sign of day of judgement here and some here.
Your second question about population increase is totally different question... I think it is better to ask separately. It has no link with first question. for a quick reference i can point to some refs here and here , will post proper answer inshallah if you ask separately.
May Allah guide us all to the right path.
Allah knows best