This is...one of the more outlandish claims I've heard.

I've not come across it in many, many books on the subject of Israel's creation. I doubt it has any historical background, and is anything but a conspiracy theory. It doesn't make sense in timeline or factual representation, either: the British abstained from the UN Partition Plan vote and were not particularly favorable towards the Jewish state idea near 1948 as they were before 1936, for example. The French, and United States, were not particularly favorable towards the idea of lobbying other states to vote yes on the UN Partition Plan (the French were reluctant to declare support and alienate Arab states), but the US agreed to provide aid in lobbying (which got France to switch by some reports) after Truman met with Weizmann, someone he was very close to. This had nothing to do with the atomic bomb in any story I've heard, it was merely symptomatic of Truman's goal of not losing the next election by alienating the Jewish vote that he believed would be crucial to winning New York (which he lost anyways, and still won the election).

Never have I heard any of this tied to the atomic bomb, even on conspiracy websites. The British, if they were so thankful, might not have abstained on the vote or might have interceded and forced the establishment of a Jewish state militarily (instead of leaving the Jews to fend for themselves). The United States could've perhaps sent troops over, if they were so thankful. In essence, no allied state undertook a clear policy of support for Israel, and even arms sales were still kept prohibited to the Jewish state by most of the Allies, while they continued to carry out deals made pre-arms embargo with the Arab states. Only the Czechoslovakians, working by most accounts as a proxy for the USSR (an Ally, but not the one your father is likely talking about) provided what was considered a tides-turning arms deal, or an arms-deal that was largely influential to the Jews winning the war, at any rate. And this was long after the atomic bomb had been used on Japan, and wasn't done until after the 1948 war began (when there had been a civil war for some months before the 1948 invasion of Arab countries started the wider war).

I don't know where this belief comes from, but it's likely just a piece of folklore or myth or conspiracy theory. I have seen nothing to back it up, and it doesn't go along with any narrative I've read that describes how Israel was established.