Ally
I have never played West Surrey. If I was down that way and looking for a game I would check into BLACKMOOR. I haven't seen it, but I always fancied a game there.
I have a few suggestions a bit further out, but not so far and as it isn't likely you will get on a club course until late morning at best the drive isn't so bad.
You could try STONEHAM. Most of the pix have been lost in the ether, but these few give a good idea of the quality - this is a vastly under-rated course on a superb piece of land.
http://golfclubatlas.com/forum/index.php/topic,30823.0.html Stoneham also features one of the great drivable par 4s in the game. A blind, slight legger left drive to a green slightly angled right to left. There is a wee bunker protecting the right (bail out side off the tee) of the green and trying to hold an approach after flying this trap is very entertaining.
http://www.stonehamgolfclub.org.uk/files/stonehamgolfclub.org.uk/35/13th.jpg While a downland rather than heathland course, HOCKLEY (near Winchester) has a lot of very good holes and I have a load of time for this place. I don't have pix, but I am very fond of this course. It climbs a hill, winds around a bit then descends. The turf is very fine and it is practically devoid of trees.
Finally and perhaps the odd chance to see a backwater links, you could try HAYLING ISLAND. Its a solid links with some charm and old world feel.
Thinking of it, Blackmoor, Stoneham, Hockley and Hayling would suit me fine for a wee trip!
Ciao