I suspect if not TR on the A32x, you’d be looking at a TR course no earlier than summer for LHR. Don’t hold me to that though. Once confirmed in the hold pool a kind soul might give you a slightly better idea. My understanding is if you only applied to LHR and didn’t tick yes to LGW, you won’t be offered LGW/EF.
Same advice I’d give to anyone applying for any other airline in the U.K. - don’t be handing in notices or making plans until you have a contract in your hand with a start date; expect a long and drawn out recruitment process with large gaps between applying, assessments and courses.
The only exception I can think of is Ryanair U.K. that want you on the line asap, and will process you very quickly. But this isn’t about them.