I suggest reading 1948, by Benny Morris.

In short, we know that US pressure was applied with carrot and stick to encourage and coerce passage of the resolution. Some of these are rumors, while others are more accepted (Liberia, for example). How widespread it was is uncertain. It's certainly possible that this was pivotal in the vote, and many suggest as much, but it did not go one way alone. Some representatives said, for example, that they were offered bribes by Arab representatives to go against their state's orders on the vote and vote "no". The choice of how to vote was influenced by politics and relations, and has been for much of history in the United Nations. The United States did keep out of the power-brokering until the last days of the vote, when it appeared to be coming down to the wire, hence the claim its influence was pivotal. It is clear as well that the Soviet Union supported partition, which ensured that the Soviet bloc (more or less, exceptions obviously occurred) went for partition as well. It was thought that Catholic countries would vote no, in Latin America, because the Vatican appeared opposed to partition, but they did not do so.

Are there any countries you're curious about in particular? I can get more specific, I just want to know which states you may want to know about.

Also, I'd be cautious about using Alison Weir as a source. She's certainly not an historian, and she has said some...questionable things in the past.