If the Holocaust had not happened, say if the Munich Conference had gone sour and the NAZIs had not had time to prepare for mass extermination...
What does happen instead? If the Munich Conference fails, and Hitler decides on war, and is overthrown by the Schwarze Kapelle...
Then not only is there no Holocaust, there is no WW II as we know it.
That has lots of other effects, including no weakening of British control over their empire (including mandates such as Palestine and their position in Egypt), continued French control of Lebanon and Syria.
British policy in Palestine could go two ways: with the Nazi regime gone, Britain may become actively hostile to a Jewish refuge there. Or, with less fear of attack, may be less inclined to placate the Arabs by excluding Jews.
There is also the question of direct influence of Nazi Germany on the Arabs; explicit meddling through the Mufti of Jerusalem, and also the effect of ex-Nazis who escaped to for instance Syria.
Another factor is Libya, which would remain an Italian colony - possibly indefinitely. It might be possible for Italian colonists to become a majority in Libya, which had only about 1M inhabitants. The conversion of an Arab and Moslem country to an extension of Europe would have a huge impact on the Arab world. (Israel may be considered such, but is politically, religiously, and linguistically separate from Europe; and in fact most Jewish Israelis are descended from Mizrahim from Asia and Africa.)
Alternatively, the war could play out largely as OTL, but the Nazis take out their anti-semitism in "mere" brutality - beatings, forced migration to ghettos, semi-starvation, murders, occasional massacres. This could effectively destroy the Jewish society of middle-eastern Europe; that is, the area between Germany and the USSR; but it would leave far more survivors. It would not be
the Holocaust.
It would also mean that the Zionist premise (Jews are severely endangered in Europe) would be confirmed (if not as strongly as OTL). Before the war, most Jews regarded anti-semitic persecution as comparable to bad weather - annoying, but not fatal, passing, not permanent, and not worth abandoning one's home of centuries over.
Thus there could be much greater migration to Palestine in 1945-1950. The Zionists might abandon the plan for partition of Palestine and go for the whole country, as needed to accommodate greater numbers. With greater manpower, they could succeed. That opens up several possibilities:
1) There is no 1948 war equivalent. The Arabs of Palestine are outnumbered from the start and don't even fight. This is unlikely, though. If it happens, no Palestinian become refugees.
2) With greater manpower, the Zionists win the war quickly, and relatively few Palestinians become refugees. Most of those return home after the war.
These two outcomes have great effects on Lebanon and Jordan, both of which were destabilized by Palestinian migration; also on the Gulf States - many Palestinian exiles got jobs there after the oil boom.
It's also possible that with the less aggrieving outcome of the war, the other Arab states make peace with Israel.
3) The Zionists win the war even more decisively, and many more Palestinians become refugees. This undermines Jordan and possibly Lebanon.
4) The Zionists win the war, but with a much larger Jewish population, agree to the return of Palestinian refugees afterward. This could also lead to peace with the Arabs.
... how differently would the Arab World have developed?
See above.
Though perhaps what you're asking is "How would Arab
thinking change if the historic mass murder program never happened?" And to that I have no answer.
One other possibility - if there is no 1948 war equivalent, or it is resolved with the least possible grief to Arabs, it may be possible that Arab countries don't persecute their Jewish populations and drive them to Israel. The continued presence of the Mizrahim in Arab countries could have lots of effects; as could the absence of the Mizrahim from Israel.
Obviously, Israel would not have been reestablished, but what else?
BTW, "re-established"? Is there really any continuity between the Herodian Tetrarchy and the State of Israel?