During my trips over there, we have pretty much been able to get on any of the courses that we want, although we always do Scotland only so I can't speak for England. I do know that Turnberry, Carnoustie, St Andrews, Troon and others that I'm leaving out were all accessible to us via our travel agent.
St Andrews was a little strange, you got into a lottery, they only allowed so many non members a day, think it was like 6 foursomes. However if they had opening or cancellations you could step in. Last trip we teed off at 4:40 which sort of ruined our playing 36 holes at each course, however we were able to play at one of the other St Andrews courses which are near by. I think this system is only for the Old Course, which BTW is my least favorite of all the courses I have played there. It was also the most expensive and had the worse caddies.