What we lack in number, we make up for in noise. ^

Train is about an hour and a half from Peterborough, IIRC, but not that frequent. At least it actually gets you to the airport, though, unlike Luton.

Minibus rental may be tricky - there are restrictions on what you can drive, under which circumstances, depending on when you got your driving licence. Sixt/ Kennings did me a good 24-hour rate for a Transit from Wembley-Gateshead, so it might be worth seeing if they do minibuses. Also check with Patterson Ford, behind Central station. That's where we used to get minibuses from (albeit not one-way).

In any case, I would guess that two medium-sized cars will be cheaper than a bus. Say £40 for the rental + £25 in petrol per car. That works out at £13 each, and you can leave Newcastle when you want.