I don't think it's anything to do with the TV Show.

"Cheers" is a very common phrase in the UK. Originally it was something you said as you raised a toast to someone. Then it evolved into something that you said as you started on a drink that someone had bought you (meaning, I guess, something like "thanks and good health"). It then further evolved into a more general "good health and goodbye" as used at the end of a conversation. That's what it means at the end of an email.

--
<http://www.dave.org.uk>

"The first rule of Perl club is you do not talk about Perl club."
-- Chip Salzenberg