If I buy an American clock, would I be able to use it (permanently) here in the UK with a travel adapter?

no.

the US uses a 60hz line frequency and the EU uses 50hz.

the clock will not keep correct time.

Yes. But while you are there, ask in the shop of they have some brains for you as well. An American one, as stupid as it can be, would certainly be an improvement for you.

Are you worried that the American clock will measure the American time (which tends indeed to be slower than the European one, if you take Apple's technology as reference)?

Update: Then why the heck do you even mention the clock, how is that relevant? If what you want to know is "whether a travel adapter for the power plug is suitable for prolonged use" then the question should have been "Is a travel adapter for the power plug suitable for prolonged use?"? Notice how in my example what you ask follows closely what you want to know?

Your problem might be indeed the tight money, my problem is your tight brain ;)

no.

An American hour actually has 100 minutes.

you know, It's like the American System (ya'll have Meters, we have yards)

but a european hour is 60 min. while an American is 100 minutes