First of all, it really depends on what the POD is for a "no-Holocaust" world. I'll assume for the sake of discussion that a certain Austrian corporal dies c. 1916, and things butterfly away from there (with Weimar Germany collapsing under the strain of depression and consecutive governments hostile or else indifferent to democracy, and a reactionary, military government assuming power in the early 1930s)...
Obviously, with no WWII, the old "heartland" of East European Jewry will not face total physical annihilation; among other things, the Yiddish language remains widely spoken as a secular everyday tongue, and many Hasidic dynasties that were wiped out IOTL continue to flourish. However, as others have pointed out, the Jews in Poland, in Lithuania, Hungary, Romania, ect. will still have to contend with the rise of authoritarian, right-wing governments that are hostile to minority rights--especially to the rights of a minority that's widely hated by most of the population, and has no "elder brother" state to act in their interests.
In the case of Poland specifically, I can see the Jews there facing increasing economic and social discrimination, from "ghetto benches" at Polish universities to state-sanctioned boycotts of Jewish businesses. Most people forget that even before World War II in OTL, Polish Jewry faced widespread pauperization due to a well-heeled campaign, centered in the Polish Catholic Church, to get Poles to stop frequenting Jewish shops.
(If Poland in this ATL gets into a losing war with the German junta over Danzig, then there's potential for widespread violence against the Jewish community. Nothing as bad as Jedwabne IOTL, but horrific nonetheless.)
The Jews of Europe will also have a number of political divisions, especially pronounced in Eastern Europe: assimilationists, Bundists (in Poland), Communists, Zionists, and the Orthodox/Hasidic communities. If Jews in Poland or Hungary or Romania face widespread violence, I could see the rise of Jewish self-defence groups (both Bundist and Zionist; don't forget, it was a Hungarian Jew who invented Krav Maga IOTL).
I expect that in the USSR, Stalin will do his best to obliterate any trace of Jewish cultural autonomy...as IOTL, I expect few Jews, barring official action, will willingly immigrate to the "Jewish Autonomous Oblast."
In Western Europe, most Jews there will continue to face the problems posed by the calls for assimilation by their community leaders. Most Jews will belong to the middle class in the West, and will vote for traditional liberal parties. I could see a number of Jews in France or Germany or the Netherlands attracted to something akin to the religious philosophy of Martin Buber or Franz Rosenweig: increased spirituality, if not outright attraction or romanticizing Jewish Orthodox traditions. Of course, many Jews in the West will probably continue, without the horrors of OTL, to view Jewish emigrants from Eastern Europe with a mixture of paternalism and contempt.
As for the prospects of a State of Israel, I believe that a Jewish state is still likely to emerge: after a certain point in the 1930s, some form of Jewish nation is probably inevitable, especially if the British don't issue something as draconian as the OTL 1939 White Paper (and with no Second World War to worry London, there probably won't be the same obsessive drive to appease Arab demands to halt Jewish immigration into the Palestinian Mandate that there was IOTL). While the Yishuv won't be getting a huge wave of German immigrants, there probably will be a couple of big "Aliyahs"--of Jews fleeing economic destitution and outright violence from the big communities of Eastern Europe. With Western European countries and the USA maintaining high immigration quotas, then Palestine will look very inviting for desperate Jewish refugees.
The final borders of any Jewish state are hard to guess. An ATL Israel might wind up with borders similar to those proposed by the Peel Commission IOTL, or larger with a better equipped and more numerous army than the IDF of OTL 1948 (a fair number of Jewish immigrants from Eastern Europe will probably have some kind of military experience).
In this ATL Israel, I could see there being a far sharper divide between the Ashkenazi and Mizrahi communities than IOTL. IOTL, Mizrahi Jews made up the majority of the State's Jewish population from about the 1960s onward. Here, with Jews continuing to arrive from Eastern Europe, this ATL Israel will have a far larger Ashkenazi community--with Yiddish emerging as a major language within the State. There will be a far larger Haredi community as well, though as IOTL, they won't be anywhere close to the majority of an ATL Israel.
(Speaking of Sephardic Jews, with no Holocaust, then the ancient Ladino-speaking communities in the Balkans won't face destruction; Ladino will be far more widely spoken than IOTL). The Sephardic/Mizrahi communities in North Africa and the Middle East will face, as IOTL, the rise of Arab nationalism and, sadly, the threats from certain quarters of state-sanctioned violence and economic destitution--especially in places like Iraq or Egypt. Besides those who move to any ATL Israel, I expect places like France or Spain to have far larger Sephardic communities than IOTL (I believe that IOTL, Spain passed a law that allows Sephardic Jews to apply for citizenship).
In the United States, the Jewish community will probably have a lot of the same issues and divisions as IOTL, although with the 1924 restrictions effective outlawing of Jewish immigration the overall numbers will remain static for a while, until those racist quotas are repealed. As in the case of Western European communities, I could see many Jews, especially younger Jews by the 1960s or so, turning against the rigorously staid and assimilationist world of their elders. As IOTL, I expect widespread Jewish participation in any civil rights movement...identification with any ATL Israel will probably be a major force in the American Jewish community, albeit slower to develop than in our world.
Anyway, just my two cents.