It's a little different in the UK- it's less official.
Here coaches are only allowed to coach skills they are qualified to, so the level a club can go to depends on it's highest qualified coach. To coach junior/senior elite you need at least a level 4 or 5 coach. Most "everyday" gyms that do grades, rec etc will have a minimum of L1 and L2 coaches, sometimes a L3.
Then of course it becomes natural that the relatively few coaches of that level will be found at the top facilities-it's very difficult to teach high level skills with no pits/strap bars/sprung floors. Then a gym/coach has one elite, and others tend to transfer in as it is seen as a successful programme.
So in England you're looking at Liverpool, Notts, Heathrow, Park Wrekin, Europa, Sapphire, Bristol Hawks, East London. That isn't fixed, and changes over time, in the 80's Ladywell and Redditch and Bromsgove were "the" clubs to be at, for example.
And would a gymmie be invited to join one based on potential or would a smaller gym refer them if they thought someone was good enough for a HPC?
This depends solely on your coach, and the gym's philosophy, with a little bit of ego thrown in. A sensible club/coach will know their limitations, and send any very talented littlies over to the nearest HPC. Many like to have a superkid to coach/in their club, and think they can get the kid to elite ranks. This is where being informed as a parent helps.
Problem in the UK is the elite track is separate, and you really need to be on it by age 7/8. By the time a parent realises a kid needs an HPC, they're usually "too old", and too behind.
There are some middle ground clubs of course, with coaches more than capable of lower level elite track, so compulsories and espoir. Personally I think if I had a kid with potential, I'd start them at one of these clubs and transfer in to a "big" club later- they tend to be high pressure, high hours, and only the really super talented are chosen, as they have hundreds to pick from.