Where in the U.K. can I get my (metallic) computer case engraved or
embossed with a custom logo?

The thin sheet metal of a computer case may not be thick enough to
allow proper engraving, and embossing would require a die to be made
up, which would be expensive for a one-off.

A signwriter would be able to turn a (vector) computer graphic into
cut vinyl fairly cheaply. Some sign vinyls can even be printed with
photo-realistic graphics.

You could also get a 'dome' badge made - these are the badges
encapsulated in a clear dome. There is sometimes a minimum run of 10
or so for these badges.

Owain

Thanks for the suggestion. I really would like the design in the metal
however, rather than on vinyl. I have a Dell computer with the dell
logo design pressed into the metal and would love to do something
similar with my other computer and my company logo.