Hi Andy,
As far as public courses go, Bethpage Black is obviously one to go for if you can get on it. If you're on your own, you shouldn't have too much trouble. It's closed on Mondays - I think that's the only thing to really look out for. The Red is also very good. It suffers from being in the shadow of the Black, but is good fun and has a lot of interesting holes on it. Ferry Point is pretty good, but it's expensive and I'm not sure it's worth the money (or the tariff on your soul).
As far as the privates go, they are private, like Queenwood or the Wisley. Basically all of them. You might be able to get lucky if you write some letters asking them for access. If you explain that you're from the UK and are visiting, you might find someone willing to let you go. Could well be worth asking the pro at your club if they could write a letter asking for courtesy. I'd say that's your best bet. Or get friendly with your colleagues!
Good luck!