The short version: the basic sizes for American beds are twin, full, queen, and king, in ascending order. The basic sizes for British beds, respectively, are single, double, king, and super-king. Single bed and double bed are understood and used in the US, but they are not precise bed sizes there. For example, in AmE I could say that a (AmE) cot/(BrE) camp bed is a 'single bed' (it fits a single person), but not that it's a 'twin bed', because twin is a particular size. Two twins make an AmE king--as one can find to one's back-breaking and love-dampening horror in hotels where they make AmE-king-size beds out of two twins and a king-size sheet. (You said king-size bed! Singular! I want my money back!!)
So, if you buy king-size fitted sheets in one country, they won't work as king-size in the other. Will the other sheets transfer? Probably not exactly.
Here's the relevant part of Wikipedia's table of sizes, with the differences between US and UK highlighted. (Australia is different in other ways--see Wikipedia for the whole story.) Note that double/full are recorded here as the same--but it's a bit tricker than that.
|39in × 75in
0.99 m × 1.91 m
|36in × 75in
0.9 m × 1.90 m
|Double/full||54in × 75in
1.37 m × 1.91 m
|54in × 75in
1.37 m × 1.91 m
|60in × 80in
1.52 m × 2.03 m
|60in × 78in
1.5 m × 2 m
Super King (UK/Ire.)
|76in × 80in
1.93 m × 2.03 m
|72in × 78in
1.83 m × 1.98 m
||72in × 84in
1.83 m × 2.13 m
Much of the rest of the story is told by these handy-dandy diagrams from Wikipedia. Here's American bed sizes (XL = extra long).
And here are the UK sizes--with an error that needs correcting: the prince size (a term I've never heard in the wild) should be small double (not small single--which is elsewhere in the diagram).
So, sheets for a US full-size should fit a UK double, but only if it's not a funny kind of double. The small double or three quarter is also advertised as a four-foot (or 4ft) bed (or sheet size) (like here).
Of course, if you buy your beds at IKEA, then that's a whole other kettle of Swedish fish. (Wikipedia has more on other countries' bed sizes as well.)
The UK bed sizes are reflected in housing descriptions. A house or (BrE + San Francisco) flat/(general AmE) apartment will be advertised, for example, with "3 double bedrooms"
or just "3 double rooms". This means 'big enough to fit a double bed'. And it often means just that: big enough to fit a double bed in, but good luck getting a (BrE/general E) bedside table/(AmE) nightstand in there.
There is a lot to say about bed linens beyond the size issues that I've approached here. But in the spirit of trying for more posts rather than longer posts, I'll save that for next time.