I do hope the OP's slightly unfair apportioning of individual blame doesn't result in our losing sight of the main issue (what a shock that would be on Flyertalk!).

I agree that it's not fair to say that "this whole thing was down to an error by a FA". Yes, the FA made a mistake in very unusual circumstances when trying to help the F pax to get off quickly, but so did the OP's mother-in-law who should be responsible for making sure she leaves the plane with all of her own belongings.

Nevertheless, the main point is that it's frankly a total disgrace if BA can't easily locate the coat. Even if I left a large item of clothing at my seat in F, I would expect the "exceptional levels of service" BA claims to offer in First would mean that it was noticed shortly after I disembarked and carefully handled and ultimately returned to me (even if I had to pay for the courier).

The fact that the OP had to speak "to people in various offices" is in itself appalling. That is not First Class service. Someone from BA should have said right away "We're sorry Madam. My name is Makeit Happen and I will personally take all the necessary steps to find your coat. You can contact me directly on 0800-xxx-yyyy if you have any questions." That is the level of service that premium hotels and airlines provide to their top customers and BA are just clueless if they don't realize that's what they should be doing too.