Each airline has different rules. Ryanair, for example, no longer accepts anything other than a passport (or, if they ever come into use in the UK, a National Identity Card). So that's all of their flights, ex Prestwick, ruled out as far as you're concerned.
easyJet (which operates flights from Glasgow to Belfast, Bristol, London Gatwick, London Luton and London Stansted) is far more flexible. They'll accept any valid form of photographic ID, such as a driving licence. (My friend recently used his Senior Citizen's bus pass as ID to travel, with easyJet, between Stansted and Belfast). easyJet also permit children under 16 to travel without any form of ID, as long as they're accompanied by an adult with appropriate ID.
BA fly from Glasgow to London City, London Gatwick and London Heathrow. To the best of my knowledge they'll only accept a passport but, since their website is unhelpful, it might be worth phoning them to check.
As far as the Irish authorities are concerned, UK citizens don't need any form of ID to enter their country. Aer Lingus is the only airline operating from Glasgow to Dublin. They'll accept any 'official' photographic ID, such as a driver's licence, bus pass or work ID. Again, accompanied children can travel without their own ID.
Chris