I was speaking to a guard at Brighton station about how stinky the coastal trains are, and he told me that Old Blue is the oldest train in the country in use today. I’m riding it to work today.
It may be touristy, but has regular scheduled operation: the Snaefell Mountain Railway on the Isle of Man.
Note that this answer was perfectly correct when written but the trains to which it refers have been replaced with the 484 series built in the mid-1970s. The link to the Wikipedia entry to which this answer refers has now been updated to reflect that.
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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:
Credit:stackoverflow.com‘