Actually the Brits didn't do this. Jews continued living in Tiberias, etc. from the middle ages under the Ottomans
The Ottomans in 1874 set up a separate district of Jerusalem and Jews escaping from Russian pogroms (note that the Russians in the years from 1772 to 1793 had participated in the partitions of the Polish-Lithuanian commonwealth. Prior to this, in the years from 1500 to 1772 the Polish-Lithuanian commonwealth,encompassing what is now Poland, lithuania, belarus, western ukraine, large parts of western Russia was home to nearly 60% of the world's Jews -- the Russians didn't have Jews before the partitions and suddenly got this big bunch of non-Russians, so they restricted them to the Pale of Settlement (essentially the parts of the Polish-Lithuanian commonwealth that Russia took)) -- anyway, the pogroms pushed Jews to America and Israel
The Jews (in Alyah I think it's called) moved in to this Ottoman province
The Jews in WWI supported Germany against Russia -- the Brits tried to get them over to their side to get American Jews to be pro-Brit
The British after WWI got this entire region plonk in their laps
Nice novelization of the Ottoman-period Yishuv is Tevye in the Holy Land, by Tzvi Fischer, available in both English and Hebrew. President Yitzchak Ben Tzvi’s major work on the subject is unfortunately rough going, even if you know Hebrew, and is unavailable in English. But there’s always Joan Peters’ Since Time Immemorial.