Zone R is the F/J check-in area in T1 at LHR. It is separate from the economy check-in, and can be used by Emerald and Sapphire status pax. The real significance is two-fold: much shorter check-in line (that part is common with the other terminals at LHR) and a dedicated security check that is much shorter, which is used by all zone R pax. Thus if your flight departs from T1 and you are flying economy, you still get the equivalent of FastTrack security (and it is typically 5 minutes vs. 30 minutes or longer for the non-Zone R security check). Now at the other terminals (T2,T3,T4) the FastTrack security is only if you are actually flying J/F and not available based on status, so even though checking in at the F desk at T4, for example, you would go to the long and slow regular security check and not be allowed to use the FastTrack check. At least that is the way it works in theory (in practice when the regular check line gets huge, over an hour long, they divert some of it to FastTrack, which then also gets to be huge and takes forever to clear .... the wonders of the BAA operations at LHR).

So using zone R generally means 15 min max check-in and clearing security time, vs. over an hour without zone R access. So it affects the timing quite a bit.

For the scenario of BA Y arrival in T1 (presuming IST arrives at T1, haven't checked this) followed by AA J departure from T3, I would say it is a wash between taking the bus and using FCC vs. going landside and walking. No FastTrack for immigration/customs (but that tends to be not bad at T1) but FastTrack for AA check-in (Park Avenue or regular) and security clearing is general fast in T3 (however FastTrack is often not operating, probably because it isn't needed that often). Either way would work within the MCT without problem. Staying airside, using the bus and FCC, is a bit easier but almost as long a walk as landside (due to the topology of T1, you basically walk over to T2 to get the bus to T3).