What can I find the oldest train in the UK still operating regularly, and the train(s) is/are operated for common service but not solely for tourist attraction?
And which route(s) are they serving?
I believe that the oldest ones in service right now are on the Island Line on the Isle of Wight, which is part of the South West Trains franchise. Not to be confused with the Isle of Wight Stream Railway, the Island Line is a regular part of the UK rail network run as normal services.
Because of the low ceiling of the Ryde Tunnel, the Island Line has a much smaller loading gauge than the rest of the UK mainline rail network, so has to use different trains. Today, those are the class 483, which are refurbished 1938 London Underground trains!
As explained in the wikipedia article on the trains
The stock is around 75 years old, making it the oldest type in Great Britain to remain in regular service
And they look like this, in their current (returned to historic) colour scheme:
It may be touristy, but has regular scheduled operation: the Snaefell Mountain Railway on the Isle of Man.