Having just sat S3 I would say the more visits you can fit in the better, it definitely helped at interview. With towers it's worth visiting some at different sized airports as well, I went to Southampton and Heathrow and they were VERY different.
I don't think NATS will have a problem with you visiting several establishments. Whether or not each facility is able to accomodate you is down to the staff at each location so it isn't like you'd be taking up lots of one person's time by going to several places or anything like that, and in my experience staff at most facilities or more than happy to show you around provided it's not too busy.