Mike,
With Ayrshire only 45 minutes down the road, not many visitors hang around long enough to sample any of the courses round Glasgow so they are all pretty well low profile but there a quite a few worth playing. Most of them date from before the war, and in a lot of cases thats WW1 not WW2.
My favs would be;
Cawder GC - a James Braid parkland course from 1934 located to the north of Glasgow. Has a second course which is fairly flat and uninteresting in places but the main course is a good fun challenge. 15-20 mins from city centre.
Ranfurly Castle GC - originally a Willie Campbell lay-out (I think) but redesigned a couple of times, lastly by Braid, a good moorland track. Located about 25 mins west of Glasgow city centre.
East Renfrewshire GC - 1920's Braid design, moorland in character. Interesting par 3's. 20 mins from city centre on south side.
Erskine GC - originally a Willie Fernie layout from 1905 (?) but subsequently had partial redesigns from both MacKenzie and Braid. Short and quirky in part with some great views over Firth of Clyde. Located 20 mins west of Glasgow cc.
Killermont (home of Glasgow GC) - not sure if you can get on here as a visitor but worth trying. 1903 layout by Old Tom Morris which has had a few tweaks by Braid but largely still the layout provided by Old Tom. Probably his best preserved layout and also the second last he did. Lots of half par holes including the dog-leg par 3 second hole. 15 mins north of Glasgow cc.
Bonnyton GC - Mike, I have to tell you I haven't played this one yet as it tends to be off the radar even for the locals but the club have recently rediscovered that the course is a Mackenzie layout which appears to have had little by way of any tampering which bodes well. Its a moorland track located 20/25 mins south of Glasgow cc.
Theres also a number of good courses in Lanarkshire (area to east of Glasgow) if you're spending any time in the area so let me know if you need any more info.
Niall