Not at all, the BMAT is sat in November and you can choose your UKCAT's date. It might be a good idea to sit the UKCAT towards the end of the summer, revising after the school year ends.
Everybody's different, there's no set amount of time you should revise. I started about a month before each exam, but that's just me (I also had the luxury of a gap year, so no school stuff to sort out). The 600Q book was helpful for the UKCAT, although the difficulties were off in a few sections. There are quite a lot of resources for the BMAT online, those were pretty helpful.