Do UK students' unions remove you as a member once you finish your programme?
Yes, because membership is conditional on you being a student - this is usually in their constitution - and they won't receive any block grant from the university for you.
If you want to remain a member, most unions/associations will offer various levels of life membership for a small fee, which will usually allow you access to the bars, sometimes the sports facilities etc, but not usually AGM attendance or voting rights.