You can use .22 semi autos for Lightweight Sporting Rifle (a discipline governed by the NSRA), and for Gallery Rifle (governed by NRA) in addition to the UKPSA bits.
Ruger 10/22 is a very common option, although if you mention it in more than a whisper you'll immediately get fanboys pouring in either slating them for poor reliability, or swearing that theirs has never misfed. Ever.
The truth is somewhere in the middle, and you probably will want to tweak the trigger and suchlike - which is where the 10/22 comes into it's own compared to other options like Browning Buckmarks and Remingtons - there are absolutely epic quantities of 3rd-party options for 10/22s - stocks, trigger kits, hammer kits, match barrels, etc. South Yorkshire Shooting Supplies (amongst others) make their own actions - so you can buy an entire 10/22 wthout a single Ruger part in it!
Obviously LSR and GR are run by NSRA/NRA affiliated clubs - look at the clubfinders on their websites for ones in your area (also e-mail them for lists if there isn't an obvious one near you. Not all clubs give permission to have their details published online, so the clubfinders are not complete and unabridged).
American NRA-style Mini rifle is not something I've really come across in the UK. No doubt it does go on, but it'll be individuals at a club getting hold of US targets and doing it themselves. AFAIK there is no organised "Mini-rifle" league, or competition circuit.
EDIT: Ah, just found this. I'd hazard a guess that any organised mini-rifle events will be run by Practical Shooting clubs, so start with UKPSA, and NRA clubs that do lots of GR rather than target rifle. They'll likely be the ones who do mini-rifle when they're not doing Gallery Rifle or Practical Shooting.
"A fear of weapons is a sign of retarded sexual and emotional maturity." Sigmund Freud
Shooting is my meditation