Nothing currently appears concerning the Cleethorpes-London service on the web sites of either local MP Martin Vickers (who is promoting the service) or the LNER (who are expected to operate it).
In its current Annual Report, BCCRP states that "Platform 4 at Cleethorpes Station is now completed, ready for the new service to London via Lincoln". However, my understanding is that the platform is primarily intended to increase capacity for the stabling of existing services.
The FBL Chairman has vowed to "eat all his hats" if the service comes to fruition in the December timetable, so I think we can assume that clearance for the service has not yet been committed.