Check-in at T1 has greatly improved since BA moved the long-distance flights from T4, so in my opinion clearing immigration/customs at T3 and walking to T1 is a viable option and should take 10 min. to clear fasttrack and 10 min. for the walk to T1 (presuming you walk briskly). This can be faster than the airside connection, plus you avoid having to deal with customs at EDI. I don't have extensive experience with regional UK customs, but once it took 2 hours and once it took 5 minutes; while LHR would be a fairly fixed time period (and you get CX baggage handling, albeit by BA on contract). It might be a good option if you have enough time at LHR for your connection (over an hour).