In time of Mohammed there were many Jewish Arabian tribes, but they didn't win out. Jesus was claimed to be the messiah, but did not fulfil the classic prophecies or ideal that the messiah would rule over Israel as a king, deliver the Jews, restore the temple, etc.
The invasions of the Caliphate established a new form of government over vast reaches of the exhausted Roman and Persian empires. Could a Jewish militant prophet unite Arabia under Jewish leadership and establish a great Arab-Jewish empire?
Of course there is no hard statistics whatsoever.
But I will take a risk and make a guess:
The whole of Arabia is 100%, I mean ALL Arabia, North and South, East and West, city dwellers and bedouins. I mean independent Arabs living outside direct rule of Byzantium and Persia.
The Jewish Arabs were from 2 to 5 per cent of the total population.
The Christian Arabs were 5-15% (all kinds of Christians).
The Zoroastrian Arabs - 1-2%.
The rest of the Arabs, the overwhelming majority were pagan, traditional Arabian beliefs.
So a Jewish militant prophet uniting all the Arabs was very improbable, almost impossible.
In fact for a few centuries Islam was a national Arabian religion - 'us' (Muslim Arabs) and 'them' (the rest of the world).
A Jewish prophet would propose 'us' (the Jews and Jewish Arabs) and 'them' (the rest of the world) - that would not work. That was the time of birth of Arabian self-identity, Arabian patriotism and solidarity. The Arabs needed the Arabian religion, Muhammad felt that and he won.
As a matter of fact in Medina Muhammad courted the Jewish Arabs for some time, the first Muslims even prayed towards Jerusalem. But something went wrong and then the direction of the prayers became - towards Mecca.
My guess is monotheism of Muhammad was closest to traditional Arabian pagan beliefs, that's why they so readily embraced it. That would take a much more time to make the Arabs Jewish. It was contrary to their traditional ways.