They took their name from a energy providing recreational drug, and were considered a one hit wonder in the US.
What is the name of this 1980's British band?

Dexy's Midnight Runners, named after Dexedrine.

http://en.wikipedia.org/wiki/Dexys_Midnight_Runners

Good answer, Jenny ! That's the band.
I was listening to a radio show that was playing good songs for a New Year's Party, and "Come on Eileen" was on the list.