Sam, are you doing this yourself, or using an overflight agent?
I have 3 URLs for these agents
Executive Jet Charter, Air Cargo & Chartered Passenger Flights - 1st Charter, UK
Overflight flight clearance specialists
FSI - Flight Service International GmbH
Never used any of them though. The nearest was when planning a trip down to Luxor HELX for which one of the above quoted me £70 for the visa, in 24hrs, whereas the Egyptian Aviation Ministry did not get me anywhere after 2-3 months of hassling them.
However, the persistent reports from people who have flown in Africa is that a bundle of US$ solves everything. It certainly would do in Egypt; the question is whether one wants to risk it because there are airports which will refuse a landing permission even when you are on final approach, if you haven't got a permit.