> Where would you put the fuse, then?

In the electrical panels, of course. You'd only need fuses in the plug if you wired up your home with a braindamaged topology like a ring circuit. (Alas, ring circuits are very common in British home wiring.)

> In the electrical panels, of course.

So how do you know what size to fit?

Or do you just get in about your panels and replace them every time you unplug a lamp to plug in a TIG welder?

You'd have a dedicated outlet on a dedicated circuit for the TIG welder. In North America, that dedicated outlet would have a different physical plug shape than your lamp, so as to avoid confusion. Of course, if in the UK you use exactly the same plug shape for both lamp and welder, I can see how that could be unsafe...

Yes, but in America you have a lot of electrical fires, and in the UK, we don't.