The main pain would be there's a security check upon entering each terminal (except, I assume, when walking between T1 and T2 when it reopens). So unless you travel when T2 reopens and you're really keen on seeing the terminal then I wouldn't bother, but if you did then you could also pop by T1 and see if they let you in the BA lounge there (although chance is very low) + without the greyhounds the lounge just isn't that good.

You will have zero luck in T4 with a BA boarding pass as QR and MH will be aware that BA no longer fly from T4.

The BA lounges in T3 are my favourite non-CCR lounges in LHR but even so are not worth the hassle and the most likely rejection.