I'd be careful of trying the off-route via Norwich option if I were you. EGNX is right on the edge of CAS and as such a handover to/from London Mil to/from EGNX Approach is easy and misses CAS. EGBB is a different story, as it is well nigh impossible to get there without having to coordinate with London Civil to cross B4/A2 (unless you want to mix it with Cessnas and Cougars at DTY at FL 45-). Then, instead of being traffic which is already in the system and heading in roughly the right direction you become a crosser to the same controller you would have been working via LAM/BKY. Take a look at the radar or TCAS picture northbound up B4/A2 and you will see why trying to cross the M1 on a friday afternoon is likely to be more successful!!
The general picture is as already highlighted: Some people will try and help you out with varying degrees of success for all the reasons already given, some will not bother. Whichever it is, just accept it. Some you win, some you lose.
Another thing: The same company sometimes files EHAM-EGCC via the CLN sector instead of via BEENO, probably to avoid restrictions in S11. The crew then ask for a shortcut on this route. It is altogether another problem due to your cruising level and the coordination sequence and will get denied on 99.5% of occasions. Forget it, and fly LAM-TNT at high level, the fuel saving will be better than low level (FL240-) via BKY.