Several Brits figured all the British Jews should go to Palestine, of whom Rev. Keith was the most famous (but there were many others). After WWI (actually, starting during WWI), the UK and France took all the Ottoman colonies outside Europe and made them UK and French colonies, and the British promised a big chunk of their colonies for UK Jews to get them out of the UK.
Not just that, but, the UK offered Central Power Jews during WWI and Axis Jews during WWII free land in Palestine if they would betray the Central Powers or the Axis. No Jews took the UK up on the offer, but the Nazis used the offer as proof that every Jew had betrayed Germany.
I've read about Jewish terrorists and Arab terrorists trying to get the UK out of Palestine during the '20s and '30s.
But the total number of Jews who went to Palestine remained small until 1948 when the US sent all the Jews in Western Europe to what had been British Palestine (but that ended in 1948 when the UN voted to divide Palestine between the Arabs and the Jews and end UK colonial status).
The US, after 1944, had open admission for Jews, so some of the European Jews shipped to Palestine managed to get US passports and moved to the US (with dual citizenship).
But after the US sent the European Jews to Palestine, many Muslim countries evicted all their Jews for stealing Palestine, and they had nowhere to go except Palestine.