Both the Luton Airport website and the one for Newquay Airport state that there are no direct flights, either scheduled or chartered, operating on this route. So, like SteveD, I'm confused!
Flying from London to Newquay, you have a choice of departing from either Stansted or Gatwick. Flights from Stansted are operated by Ryanair. Their policy on acceptable forms of photo ID is given here:
http://www.ryanair.com/site/EN/faqs.php?sect=p id&quest=photoid
Flights from Gatwick are operated by Air SouthWest. Their far more liberal rules are quoted here:
http://www.airsouthwestextras.co.uk/faq/showfa qs.php?id=33
Chris