I know Royal Blackheath do corporate days. Not sure of the cost but should be good as a 'nice' club. Claims to be the oldest Golf Club in the country, so a bit of history there too.
There will be an Old Etonian Golfing society. Pull the right strings and you can get on anywhere (possibly at a great price), not that money is necessarily an issue.
Sunningdale (Old), Walton Heath or West Hill would be my choices for the London area.
Just noticed you mention a weekend morning. That might rule out quite a lot of member clubs. My club you can only play with a member at the weekend, and strictly no societies.
Also some clubs like Sunningdale have handicap limits of 18.
I know Royal Blackheath do corporate days. Not sure of the cost but should be good as a 'nice' club. Claims to be the oldest Golf Club in the country, so a bit of history there too.