http://www.golfclubatlas.com/forum/index.php/topic,64891.msg1546751.html#msg1546751is the previous GCA link on Flempton, which does make a super pairing with The Sacred Nine for a day's golf in rural England.
This is generally the driest part of the UK, and has been exceptionally so this summer: the Old Course at Gog Magog has certainly returned to its fast, running, chalk downland origins, and now would be the time to play it if you want to experience the ground game inland...
New Zealand likewise comes hugely recommended, although I have to add my perennial warning to all American GCA friends wanting to play golf around London: the traffic around the M25 (London's beltway) can be horrendous at times, and it almost always takes much, much longer to drive even twenty miles than GoogleMaps might suggest.
Both New Zealand and RW&NGC are smashing examples of quiet, old-school (2-ball) British golf clubs. To that listing I would add both Woking and Huntercombe and, a bit further away towards the Suffolk coast, Aldeburgh. All have been profiled on this site. Anyway, enjoy!