I have started a couple of pilots in the very first (convoy, I think it was) campaign. What are the time periods of the different campaigns and when I finish the convoy portion does it automatically go to the next time period?

The game follows very closely to the RAF Museums info on the Battle of Britain :
Game RAF Museum
Convoy Battles 10 July 1940 Phase 1 26 June-16 July 1940
Coastal Battles 08 August 1940 Phase 2 17 July - 12 August 1940
Assault on Fighter Command 24 August 1940 Phase 3 13 Aug - 6 Sept 1940
Daylight Attacks on London 07 Sept 1940 Phase 4 07 Sept - 2 Oct 1940
Fighter Bomber Attacks 01 Oct 1940 Phase 5 03 Oct - 31 Oct 1940

Yes, that's about correct. If you go to create a pilot, you can go through dates and see the periods. It continues yes, provided that your squadron did participate you will carry on. Some were pulled back etc so you'll do patrols from further back in that case unless you join another squadron.

That's nicely done. Very realistic.