BA do not approve taxying out with engine(s) shutdown, although if you taxy out to a remote holding area awaiting a slot, which could be an hour away, it is then usual to shutdown all engines until about 25-20 minutes before the slot time.
Taxying in, the procedure is available to shut down an engine at pilot discretion if desired in order to save fuel. This applies to 737, A319/32/21 and 747 aircraft (I can't comment on 757/67/77 aircraft). On a shorthaul twin, I believe the saving is about 20kg of fuel per sector, which in itself is not much, but with about 110 SH aircraft doing about 6 sectors per day that equates to about 13 tonnes per day @ about $650 per tonne.
Circumstances which would preclude shutting an engine down would be local restrictions eg T4, local gradients, very short taxi time, etc.