I have had a look on the Great Park website. There is a section on there about previous changes to services and may give a clue about funding (obviously the service info is obsolete)

'From Sunday 30th May the network of bus services to NGP was significantly enhanced. The changes include the introduction of a direct link to the City Centre on the new X40 Park and Ride service and the introduction of service 46 between the City Centre and Featherstone Grove. These changes have been introduced in response to requests from local residents and to provide a new service to the new Neighbourhood Centre to the south east of the SAGE building. The service improvements have been funded jointly by the NGP Consortium and Newcastle City Council, who continue to work together to deliver high quality sustainable transport links to and within the NGP development as part of the Travel Plan Strategy. The Travel Plan Strategy sets out the commitment of the NGP Consortium to delivering a wide range of transport projects over the coming months and years, from bus services to cycle paths and lots in between. We'll keep you updated on future plans and improvements and for more information on the new bus services please visit www.nexus.org.uk/bus or email rob.bettison@travelplanservices.co.uk'

Having looked into who the NGP partnership are, these are the developers i.e Persimmion, Barratt. Now either they are funding the service from their own pockets along with Newcastle City Council, or residents pay some sort of service charge, which contributes towards the service.