In the first place, by the UN's estimate there were not "a few hundred thousand" Jews living under the British Mandate in 46, but over 600,000 in 1945. Second, the collapse of the British mandate had really nothing to do with the Shoah, and everything to do with their fragile economy post WWII(or do you believe that they quit the Raj as well because of Treblinka) so it is fantasy to imagine that the Empire could have held on the in the Mid East "for many more years." Lastly, my figure of 3-4 million is based on a pretty straight forward calculation. The 3 million Jews in Poland tried desperately to leave emigrate even before the Nazi invasion -- including to the Mandate -- but were denied access. Likewise, Soviet Jews were also trying to emigrate with the same results. Since no one would take them before the Shoah (or likewise) absent the Shoah, it is reasonable to assume they would have gone to the one place that would have had them, Israel.
U.S. asks Uzbekistan to join anti-ISIS coalition (Reuters)
from the article: It's wrong to suggest Israel was direct result of Holocaust