Gonzo, I would say that relocation to a different ATC Unit and your issue are two different subjects.
As I understand it, you are being made to travel longer within the airport to get to work.
OK, this is not a great situation but I do not believe that you should be compensated. You are not, after all, on shift during this extra time. This is just that your working environment has changed. You either accept it, as controllers accept change regularly, or find that the situation is unacceptable for you and look for another post. If enough EGLL TWR Controllers expressed their wished to leave, then you might have something happen.
I would push your management to provide you with the time that they require within the airport to get to work - such that if the airport were not transporting you to your place of work within the prescribed time, then it is not your fault! This however could be problematic for logistic reasons.
I assume that you have already approached your direct managers and requested some assistance with this matter, if not, then try that. (Free coffee machine maybe??)
If the union believe you have a case to argue, then so be it, but I fully understand all the other controllers having a problem with you requesting more money for this.
A small example - the staff carpark that was used, is to be built on with an airport building - another staff carpark is allocated but involves longer walking distance to work - should all airport employees be compensated for this?
Not as extreme an example as yours but what do you think?