Just to clarify some of the posts above for anyone else looking at this in the future

T1 is a long straggling terminal, with the domestic wing at one end and Flight Connections at the other (more or less). HEX is nearer the domestic end. Old BA hands will recall that long elevated walkway between the international and what was the main domestic area, it's still there but looks a bit different now.

Therefore if arriving at T5 and going to T1 (and vice versa), there is sometimes a case for going landside on HEX if you are domestic, to save a long hike within T1. I personally go airside since on my timings it is still quicker, mainly because landside HEX runs every 15 minutes and the airside bus is every 7 minutes. Other travellers will have other (valid) experiences.

The case for going landside would be even stronger if going T5 Domestic to T1 Domestic (and frankly mandatory if you don't have your passport on you). The layout is better in that scenario. T1 domestic to T5 domestic is less clear cut, but again if travelling without passport you must go landside.

If going T5 international to T1 international on flight connections then going airside is best (in my opinion) since there should be less of a hike and you avoid a passport check.

If you have checked baggage to pick up you must go landside.

If you arrive airside in T1 or T5, you still have the option to exit to landside when the bus gets you to this terminal, instead of going via Flight Connections.