I've been looking at how the French handled Algeria and Tunisia, and specifically the Jewish populations there. The sentiment seems mostly to have been, "We don't particularly like them, but at least they're white!" Similarly, the French have none of the pro-Arab feelings that the British have, so I strongly suspect we'll see unrestrained Jewish immigration to the region...probably coupled with some Arab immigration from Syria and Lebanon, but not all together that much.
The main problem is that the vast majority of existing Zionist settlements at the time exist outside of the designated French zone, and so the question becomes whether the Jews already there will migrate into the French region, or whether Zionism will continue apace in both. The British may possibly just declare the entire Mandate of Palestine (or whatever) to be part of the new Hashemite emirate (which will probably be called Jordan as be OTL, but could well take the title Palestine. In either case, less emphasis on Jewish immigration to the region will result in less legislation against it - I suspect there will be modest Jewish immigration even to the British parts of the region (notably Tel Aviv, the Jezrael Valley, and the highlands just south of Haifa where all the oldest Zionist towns are).
In the inter-war period, the French are still going to love Lebanon with its Christians, but they might be even more neglectful to Syria, funneling some development into Palestine/Jerusalem. At any rate, the French Rothschilds are probably going to be even more involved with financing and supporting developing in the region - look to Ashdod or Ashkelon to become a major port city, and Jerusalem to become a European style, Jewish majority city by 1940.
During the War, P/J will definitely be either Free French or immediately occupied by the British - the leaders are probably smart enough to go the first way. On the other hand, the terrain means that a wily Vichy commander could probably keep the British out of Jerusalem for quite some time if he put his mind to it (Jerusalem is completely surrounded by surprisingly high hills, and there's basically only one major road in or out of the city).
The region will probably gain independence shortly after WWII, and I suspect it'll carry a government that somewhat resembles the trainwreck that is Lebanon, with quota systems and particular offices reserved for members of this or that community - I suspect Arabs, Ashkenazi Jews, and Sephardi/Mizrahi Jews. The Bedouin might also get guaranteed representation, but I strongly doubt it. Where it goes from there depends on the precise demographic breakdown and how it evolves.
As for the dispensation of the British part...that depends completely on whether they give the whole thing to the emir (more likely IMO) or be sensitive to the fact that the region west of the Jordan is very Jewish. Something tells me that the socialist Zionists aren't going to want to be under an Arab monarchy.
