The Lincs AIAA and Vale of York AIAA are my local area. Certainly at weekends, no problem with transit whatsoever.
Durham TV are quite helpful (Class D) so should allow you transit N to S. Newcastle Approach will normally give a handover.
South of Durham TV, Leeming are usually closed (so no FIS/ RIS); ditto Linton. Leeds Approach gives the best service for the first part South of the Durham zone, then freecall Humberside for the service.
The Vale of York is often quite busy with gliders (on a nice day).
Where you go next depands on if you are E or W of the Lincs bases (Wadders/ Cranwell).
Waddington will provide a good zone service during op hours (see NOTAM). Note Cranwell is usually active at weekends and Waddington is always active with AWACS so you need to be talking to them in the AIAA, especially close in to the fields.
Waddington will provide the service over towards Wittering (Cottesmore closed at wekends). Thereafter freecall Cambridge before going to Standsted.
As noted, they will provide a service (Essex radar) but becareful to keep clear on controlled airspace. They are still busy at weekends thanks to Ryanair et al.
Good flight.
h-r
NB, sorry for the spelling errors, typed in a hurry at the end of the day.