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.