This isn't an area of personal knowledge for me, but having watched the threads on here there seems to be up to three approaches:

1) If travelling First, very high value passengers, or there is a genuine emergency (e.g. medical equipment), then BA are willing to initiate searches to track down missing bags. We are talking well under 1% of incidents here.

2) Otherwise you can spend hours on the telephone and hassle the system to getting your bags. It probably does speed up the process by 24 hours (max).

3) Do nothing / the minimum. The bags will show up eventually.

BA provide resources for point 3, but will on outbound journeys be generous to replace items genuinely needed. Insurance covers the rest, and they ping BA heavily as needed. I am not saying whether this is right or wrong, but approach 2 is rarely going to make much difference. For an easier life - not that it ever affects me - I'd personally ride with approach 3 and send in the bills to BA / insurers.