I suspect that this will lead to Judaism being much less of a "special" religion, so to speak. With - as you note - a Sadducee leadership and a well-defined national center, I think it'll end up as a relatively minor national religion. Combine that with the wave of religious unrest that is going to affect the Roman Empire sooner or later (Christianity or no), and Judaism seems likely to be a peculiar, but nonetheless thoroughly integrated member of a pan-Mediterranean religious system - comparable to Lingayatism in Hinduism. Jews are still useful middlemen in Mesopotamia and Arabia, but they don't have any special reason to move en masse like they did IOTL - and they'll always feel an allegiance to Jerusalem and the Temple.
Christianity is screwed, at least for Paul's gentile outreach plan. A more confident and cohesive Jerusalem is going to be a focal point for the cult it spawned, and no matter how hard Peter or Paul try, Ebionism is going to be the most natural conclusion. Consider also how much less authority Christianity will have, among Jews or Romans, without a little Apocalypse happening right after its appearance. No reason to discard the idea of a prophetic leader and deified mankind in general (c.f. Buddhism), but it will probably come from a Hellenic or Persian mindset rather than an Abrahamic one - and therefore much less anxiety about idolatry and polytheism.
Islam is, of course, butterflied away. The Arabian Peninsula is still going to have an economic crisis around the 6th-8th centuries, but that doesn't necessitate a political-religious movement. More likely is a time of massive infighting, maybe an Arabian conquest of nearby areas, but nothing anywhere close to the scale we saw OTL.
I love these kind of WIs.