Sorry but that's all I know about transferring from EOG to LHR. I haven't done it myself so my informations are more what I gathered from others who have been there and done that. Best you can do is:
-Talk to one of the union (or both) to find out exactly what are your options.
-Be patient because in the airline industry all is down to seniority so it's only a matter of time before you get your own choice.
Last piece of advice for tonight. Re-think about what you expect from the job, maybe (just a maybe), you'd rather do something else while still remaining in the airline business. There is nothing wrong with it, some of my friends have chosen to do that and they are much happier like that (while still enjoying the bits and bobs the airline can offer, including further/faster promotion). Cheers