Disclaimer: this is Alternate History and it does not represent the real world.
The point of Divergence is that Britain is not the only one managing the Mandate of Palestine. Italy and Poland also managed their own division of Palestine. Poland later created Israel in 1938 and gained full independence in 1945. Britain had created the State, later the Arab Republic of Palestine, in 1946. And the Kingdom of Jerusalem? It was Italian Palestine. After all, the Savoy claimed the title King of Jerusalem. It was also useful as a penal colony. But later, Mussolini declared the independence of the Kingdom with one of the sons of the King of Italy. Surprisingly, the King joined the Allies, enraging Mussolini and one of the reasons why he embarked on the Egyptian campaign that is a failure as in OTL. There are more Jewish refugees in Israel because Israel is created already for them. It was also the home of the Polish government-in-exile until 1989, when it joined Walesa's government back home.
Palestine is a little better than in OTL. The bad news is after the Holy Land wars, Palestine is still the poorest of the Holy Land states, with the Latin Kingdom being almost as par with Israel's standards, similar to OTL. The official languages of LKJ are Latin, Italian, French, Arabic, Hebrew, and Greek. Of course, it was a favorite target of ire of American Protestant fundamentalists, but the LKJ had very good relations with Israel. The Latin Jerusalemites had poor relations with Palestine, and of course, the Israeli-Palestinian relationship is as in OTL.
Blue Is Israel, Red is the Kingdom, the green is Palestinian. BTW, Poland's government in exile fled to Israel precisely because it was the last territory Poland has left.