Jack
My take on it is that you should be fine. At LGW, GB are handled by Aviance, so check in and baggage handling will work. Catering is not BA, neither are Flight/Cabin Crew.
At present BASSA is balloting BA mainlline Cabin Crew for a strike mandate, whether this is given remains to be seen, we will know in Mid January. There are no plans at present to ballot BA pilots for industrial action.
Any BA shutdown due to IA is VERY unlikely to see GB or Bmed operating any mainline BA services, they (BA/Bmed) would protect their own network first, and secondly the number of mainline services they could operate would be such a small percentage as to make it pointless, and more importantly the cash cow in BA is the longhaul Atlantic route network which they couldn't serve anyway.