As far as I know there's just one major player in Glasgow, the TOA, and they have a bit on their site about a knowledge school:
http://www.gwtoa.co.uk/taxi_school_training.html
In other cities individual operators may advertise in the local evening paper for drivers, whether or not they are attached to a large office, but things may be different in Glasgow.
I doubt if Glasgow and Edinburgh are hugely different, but looking at the numbers Glasgow has a large private hire sector, but Edinburgh's is relatively small. It may be easier to get into PH than taxis, particulary if the taxi knowledge test is quite stiff, especially as you seem to be just looking for part time work - PH in Edinburgh don't need to pass a knowlege test as far as I know, but I don't know about Glasgow.
Dusty