Your challenge, if you choose to accept it, is to make it so that British Astronauts land on the moon, and explain the ramifications of such a landing.
You probably need a very different world, perhaps no world wars or even an America that became a dominion like Canada did.
A joint project between the likes of Australia, Canada, Britain (With no world wars) might be an option as well.
I think the ramifications of landing there are dwarfed by the fact that world is clearly very different already. I would like to think it would reflect a world with more focus on exploration and great projects rather than war. Britain for historical reasons might be more interested in settling the Solar system than NASA.