My 'starter for 10'! Stand by for a bit of very sad geekery!
According to to the 1977 Guiness Book of Air Facts and Feats:
The oldest airline service still operated by the original airline company was the KLM London/Amsterdam service started on 17 May 1920 using a DH16 of Aircraft Transport and Travel piloted by H 'Jerry' Shaw. After that, all early KLM flights were operated by the same airline under charter.
Now the interesting bit ... that service still operates - Flight KLM 1000 Heathrow to Schiphol. That could be a contender!
Right - I'm off to count my collection of used rivets ... berble, berble, berble ...!
Regards
DS