I've got hot water coming from cold AND hot water tap at a sink in a laundromat.
I saw some posts that suggest it may be related to washers?

Something about a mixing valve in the open faucet? We have 20 washers here.
They're all connected to a garden hose type connection.
The faucets are left open and the washer itself turns the water on/off.
Where would this mixing valve be and how to fix it?

No recent work/adjustment has been done to cause this.
It seemed to happen on it's own.
Both pipes coming from the gas hot water heater are hot to touch.

What can I do?

Both pipes coming from the gas hot water heater are hot to touch.

That is not supposed to be hot on both pipes. I suggest you call for service and get the hot water heater situation squared away first.

The water heater is not that old, also its connected with an old boiler(?).
The 2 connecting pipes have thermometers at 130F,
And are connected with the 2 bottom heater pipes (inlet/outlet)
There is also some occasional leakage from the overflow(?) pipe which may be normal?

The bottom outlet pipe is slightly less hot than the top inlet pipe.
The one coming from the very top of the heater says 270F (almost top of the scale!).
The cold water tap in the sink upstairs doesn't run hot water anymore,
But whatever it was, it may be a warning of a problem in the future.
What do you experts think? Before a local plumber finds an 'expensive' problem.

It looks like your new water heater is just a stack storage tank.

There is also some occasional leakage from the overflow(?) pipe which may be normal?

No not normal

The one coming from the very top of the heater says 270F (almost top of the scale!).

If that is reading 270 degrees shut down the system and call the plumber right away.

Thanks will do!