The exact climate of 250m years into the future can't be verified but there would definitely be a lot of deserts and hyper continental climates. Mountain ranges will probably act as massive borders between regions with different sides of those ranges having extremely different climates.
and the mountains themselves having very interesting climates. That's where all the fun is to be had!
Oh yes of course! Those mountains will probbly have some kind of very wet (if bordering the massive global ocean) or very dry climates. Not just that but due to high elevation / steep valleys loads of micro climates. It is interesting..
The UK would be uninhabitable. I guess most of the hypothetical population would live on the narrow coastal strip on the west coast of the Americas, around the Indian Ocean, and along the coast between Australia and Asia.