I had always lived in the belief that a 15W soldering iron ran cooler
than a 35W iron but recently I read something which said they both
run at the same temp.

Presumably the real difference is the speed at which the iron reheats
once the thermostat kicks in to say the iron is too cool.

I'm in the Uk and I currently use a 25W Antex. It seems to "run out
of heat" all too often when I am soldering undemanding things like a
fine wires to small tag.

If the temp is the same then would I be ok using something like a 35W
iron even for working on electronics components.