Schuko is rated for 16A continuous for one hour, 3680W, I think the UK plug would actually take quite a bit more you're just being conservative. Or something odd about ring circuits I don't want to think about.

In any case practically nothing in a household actually uses 3kW. A stove, yes, but that's connected to three phases and without a plug (usually 3x20A over here -- CEE plugs can do that but they're chonkers and how often do you move your stove). Newer dryers should stay under 1kW, the standard high load appliances are kettles and hair dryers.