Asalamu 3alykum xsilentmurmurx
Your question to me seems very difficult to answer.
Insha'Allah I may add some of my views and maybe it will help in some way.
Lets say in our era today a book was being spread and it claims to be from God. Lets say this book has been spreading rapidly on the internet or through people.
How does a Jew, Christian or Muslim cope with this book while this book is completely different from those messages?
The way I can see any one leaving a religion or path is if one is given a better position or manifestation of truth of what is already with them or if they are presented with clear evidence of their falsehood.
The Quran for example seems to demand from those who are searching for evidence that to present a book then that is a better guidance than the Quran and what was given to Moses which is from the Creator so that one can follow it? (28:49). If one can't then they are just following their own desires.
This seems to indicate that the Quran is holding its position and also challenges that there is no other better guide and is expecting humans to follow the best of what they are presented with and also seems to indicate that there is a guide from the Creator.
That being said
There seems to be a universal truth in this life. Its sometimes a new message to a people and sometimes it is a reminder to a people who were guided before. This universal truth has always been from the same One True God. Although it comes in different forms, the root of the truth is always from God and is always manifested somehow someway.
One is expected to leave a position in belief if something seems false or a truer position has come to light.
The guidance of God has always seemed to do just that. It brought people from darkness to light whether it was a community who was guided but lost somethings of guidance or whether it was a new community.
So anyone presented with a true prophet or messenger would always recognize them by the words that they speak. Sometimes they expose the falsehood and sometimes they confirm the truth.
Now to answer your question, the Jews and the Christians assuming they are a people who claim to uphold truth, they should also recognize truth when it is revealed because it would be a confirmation of what is with them and they are familiar with the messages and voices of the Lord. They were people who would recognize the words of God better than they would recognize their own sons.
Why is that? Because they are familiar with the chains of messengers and prophets that preached the same religion. Prophet Muhammad is no different. He also was sent to them with the same religion.
Therefore there was an expectation that once the message was manifested they should also believe it was from God .
The indicator is that truth itself stand outs. It is recognized.
For example,
Although many can believe in prophets and messengers because of what they here about them, some would also doubt that some messengers and prophets were actually real. They would need evidence or some manifestation of there authenticity.
Some also might believe in prophet Muhammad as being a true prophet just from heresy but some also do not believe in the prophet Muhammad as being a true prophet because of what was said of him from traditions and heresy, but they would believe in his authenticity as a true prophet because of the mighty words the Quran contains. It is the Quran that manifest that he was a true prophet of God and not just a man claiming to be so. We realize after reading the Quran this fact. Many come to this conclusion through so many ways from the Quran.
You said
how would the Jews and Christians be able to distinguish between which Prophet is the true Prophet and which one is a false one? There has to be some indicator in the Bible that would set apart our Prophet from any false claimants during his time or after his time. Or else, how would anyone be able to tell who is speaking the truth?
I think this question applies to us just as much it applies to them.
How do you tell that prophet Muhammad was a true prophet rather than just one who claimed to be one and was believed by many people especially when one does not believe in the heresy of people and wants evidence?
What makes you so sure that he was not a false prophet? We did not see him nor do we have any physical indicators.
We did not see the miracles that were given to prophet Moses, or Jesus, but we believe in them. why?
Those are my thoughts
Salam