Haven't actually stayed at the Hotel but have parked in the long stay car park at Newcastle 3 times without any problems.
The Brittania looks as if it has its own car park, but i would imagine you would use the long stay at the airport. A free bus runs contantly from the long stay car to the terminal building and there is also a phone on each bus stop to call it.
On departure you go back to your car and then drive to the security office to either pay or present both your parking ticket for validation along with your car parking voucher from whoever you prepaid your car parking with.