First, I think that a USSR-leaning Israel, whether outright Communist (very likely, by the way, in such a scenario) or simply socialist, would definitely involve almost all the Arab countries going to the US, if only by default. Which may delay or outright prevent the nationalization of the Suez Canal - after all, whatever probably-military dictatorship takes over in Egypt won't possibly be socialist-leaning, and so they'll be much less likely to nationalize anything, though we may see the UK leaned upon heavily to sell the Canal at a ridiculously low price to an Egyptian company or government. With US-backed Syria right there, Lebanon may be less enthusiastic about inviting in the Marines when their religious conflict inevitably reaches a hard boil, though what comes next is hard to say - bloody civil war in the 50s? Syrian annexation, in part or in whole? Soviet/American proxy war in Lebanon, by way of Israel and Syria? Certainly the coolest possibility is a militantly atheist Communist regime, but I'm not sure where such a thing could come from, except by way of de facto Soviet/Israeli occupation.
Domestically, American Jews will tread much more lightly. As previously mentioned, American Jews were incredibly involved with the Civil Rights movement, as well as being highly involved in all of the shades of the American Left, from Democrats to the Communist Party of the USA. With a Communist Israel, all American Jews will be under suspicion from the FBI, even the ones that openly revile the horribly misled state. It might lead to an even leftier Jewish population, actually, as Communist Israel starts emphasizing the Socialist currents in recent Jewish thought and tries to unify Jewish culture and Communism. This, in turn, may hurt Jewish assimilation - after all, these weird Commies are seeming less and less like civilized white folk by the day, no matter the color of their skin. Antisemitism will probably be more socially acceptable, especially amongst the rabid nationalist anti-Commie crowd (in fact, anti-Jewish slurs may be generalized to Communists in general). While I don't think that Jews will ever come to be regarded as a Fifth Column (no matter how much the Mossad actually does end up recruiting them...), they may end up receiving the same kind of suspicion from "the Establishment" that blacks often did in the 60s and 70s.
A related, equally interesting question: what effect does this have on Jews in the Warsaw Pact, of which there are also quite a few?