The British authorities absolutely won't care when you left home. They care about when you arrive in their country.
The airline's check-in staff should not care about things the UK authorities don't care about. In principle they may be badly trained and think it matters whether the visa is valid already when you check in. This is not particularly likely, and there's not anything you can do in advance to guard 100% against such incompetence.
It might possibly help if you can say at check in that you phoned the airline's customer service ahead of time and was told that it's OK -- at least in the sense of convincing an incompetent agent to check with their superiors instead of rejecting you out of hand.
But all in all I don't think it is worth worrying unduly about.