Someone arriving at T1 would go through passport control at T1 and end up at the baggage area T1. There definitely was and might still be a connections path that led from the baggage area back up to security which cut the queue short and saved some walking.

Arriving in T2 is a different, much larger, immigration hall and there's a dedicated connections path starting with immigration. it splits in two with one for T1 so watch the signs. Though I'd you ends up in T2 they're both connected airside anyways.

The BA flights usually leave from the 300 gates which is close to T2 but occasionally from the gates that are really far away at the other end. Usually that's the first flight of the day that does that I think. If you leave from those it's more like 20 minutes walking.