Any items left on the aircraft at LGW are turned into BA's lost property department (NOT BAA - though LGW is run by a different company). I suspect this is what has happened to your mother's coat.

I am unsure of a phone number which can be used, but perhaps you could phone BA and ask for one. You'll be amazed the amount of unclaimed items which are in our lost property at LGW.

And, I agree it's unfair to put it down solely as an error by the crew. As you say they had a medical emergency on their hands and that will have been their primary consideration at that time. It's unfortunate that your coat was not given back to you, but somewhat understandable under the circumstances.