We have three towel radiators, all of them have two tap like fixtures on them. We tried turning these, first one way, then the other. Wouldn't turn off. Ended up turning them off at the wall switch. Then they were still on, so we unscrewed a panel and took a fuse out, still on. I can't remember how we turned the bloody things off in the end. I think a plumber came for something else and knew which way each of the taps needing turning, maybe.
Turns out they operate on both oil OR electricity. We didn't have any oil at the time! They were also costing 10 kWh each in electricity to run.
They were on when the heating was not, so yes they are connected to the hot water.
My LL mostly employed cowboys to build this house, I don't suppose that's the way towel radiators are meant to work, but I am sure piglet John can confirm.
The tap temp will be down to the temp it is set on the hot water thermostat, that can be turned down.
I also notice ours gets far too hot if we don't use all the hot water in a day, or if the house has got particularly hot (probably the loft) so I guess the pipes get hotter?
No I don't think you need an emergency plumber.