It is my understanding the first "Jazz" service was the high frequency suburban services from Liverpool Street that ran to places such as Chingford, Enfield, Hertford East, etc. It started in the 1920s when Jazz was the latest "in thing" .
Edit-- Buttsy beat me to the answer by 2 minutes
Yes, Jazz was the latest "in thing" I understand this was not the main reason why the service was so nicknamed.
I gather this was because to speed-up boarding, the coaches were colour-coded. I'm indebted here to Jim Hill, and his book "Buckjumpers, Gobblers & Clauds" [Bradford Barton 1981 - ISBN 0 85153 396 5], where he says; "A broad band of paint of appropriate colour was added to the long panel just beneath the roofing - yellow for 3rd class, blue for 2nd, and 1st was left alone - hence the nickname from these jazzy colours" p.52].
So, Buttsy wins by 2 minutes.
Edit:- Oops - whilst I was giving my lengthy explanation, I gather Buttsy has declared open floor!