Current research is that it would get a lot drier than it is currently. Also the AMOC collapse process would play out over many decades (~50-100 years) it will not be a sudden event.

Warm oceans, as they are, would surely bring rain? Hence why the UK is so rainy. The ocean is warmer than the land, it's like warm water in a cold room, you get the moisture coming off it. If the ocean isn't warm any more, I reckon (as an amateur engineer not climatologist) that would reduce the rain we get.

That is one part, the other part is that you would get more persistent high pressure blocking patterns over Europe due to reordered atmosphere circulation which will lead to more intense droughts. That is what happened this summer.