I'm based in Basingstoke and joined TFSA (
http://www.tfsa.co.uk/) as they shoot on an indoor range for small bore and gallery and then have regular club trips to Bisley. I don't know if there's any spaces available right now as the club is pretty full but you can always ask.
The other place I considered was Itchen Valley (
http://www.ivsc.co.uk/) but I see the web site says they're full too although you might get on a waiting list. Probably worth asking.
Sorry that's not as helpful as it could be but you might get lucky. I certainly don't know of any other clubs that shoot lots of different disciplines so I think other than these the NRA club is probably your best bet for getting started. You could probably find a small bore club local to you in addition to the NRA full bore days if you went down that route.