Keeping the British away from nuclear reactors is pretty simple, and doesn't even need much of a POD because it came down to one man - Louis Mountbatten. He went over to the US and charmed the pants off Rickover, who up to that point had been really hostile to the UK getting US reactor technology. No Mountbatten as First Sea Lord (not difficult - half the crew of Kelly were killed off Crete when she was sunk, and he was associated with various political disasters such as Dieppe and the partition of India which Churchill was mad at him for for some years) means nobody capable of charming Rickover into getting the technology handed over.
How did Mountbatten manage to pull that off with Rickover? That's a story in its own right.
The Euratom agreement is unlikely to be affected - that means the French go for PWRs for civil power on schedule anyway. It didn't affect the British much, so they probably continue with the Magnox/AGR route.
As already mentioned, the British had already decided that a naval reactor needed to be a PWR for various reasons. Developing one is within their competence, although it will have a knock-on effect on other projects due to a lack of staff. Best guess is that MAGNOX continues a bit longer as a programme, and when AGR comes along it is a clearer evolution of/closer relation to MAGNOX than it was in OTL.
I'm wondering more if the British would adopt water cooling earlier due to its development for the military program, or perhaps join Euratom and potentially push for commercial gas cooled reactors. At the time many were impressed with the British and French gas cooled reactors, and they actually led water cooled reactors in terms of number of operating hours achieved for several years (I think until the 1970s).
Apart from naval reactors, the British didn't really push much for light water designs, so they could have an interesting impact on the development of nuclear energy if they have to go it alone for naval reactors and/or join Euratom. The
1958 Euratom Cooperation Act represents the point at which the United States flipped most of Europe to light water (specifically PWR), and it's possible earlier British involvement could lead to a different outcome. In 1958 the British were the experts in nuclear energy, not the United States, so the British stance could have significant weight.
Cooperation between the British and French is problematic - the 1958 US-UK Mutual Defence Agreement rather restricts what the UK can share in nuclear terms, and the UK interpretation of it was rather paranoid in that if in doubt they wouldn't talk about things. An additional cooperation is that the 1958 act gave the UK access to highly enriched Uranium of which their own supplies were inadequate - and which is required for a submarine reactor of reasonable size and lifetime. Not impossible - the French have built one capable of using low-enrichment fuel - but the performance is very much inferior to one using weapons-grade fuel.
A French agreement would be an alternative to one with the United States, not one that would take place at the same time.