I do have sympathy for you but I think you're nitpicking a lot. I'm a survivor myself and know Fentanyl very well. I know the pains and exhaustion.
With all the online "unable to check in" notices did you try calling as this is so important?
As to the wheelchair service, why didn't your wife and daughter sit at one point and you go handle getting the chair from 1 person instead of her walking around?
I'm glad that longest flights worked out for them as sitting that long must have been intensive.
Flight delays happen and buffers for people that are ill should be longer than standard. This is a learning experience. The fact that the gate was far was not BA being evil and laughing sinisterly at your wife. Gates do close at a certain time ahead of the door clothing. Your Wife and Daughter not being in the Terminal at that time would allow them to lock in the flight and get it out early.
The Wheelchairs at LHR especially the guy getting lost is idiotic of that worker and a legit complaint. They should know their way around and be able to assist people in a respectable amount of time
As to your daughter having to purchase water...they were in a lounge without water?

Not trying to say the experience wasn't ideal but maybe it was made worse by not taking the appropriate actions ahead of time. Yes you did note she needs a wheelchair but it doesn't seem like you left enough time between flights through LHR, wife should travel with 2 days of medication at all times (this is a rule I have for myself whenever I travel. In fact i my pocket I have 3 days of my meds as I'm on all oral meds) and contacting them when online check in failed. Maybe you did the last one but you didn't say you did.

Write your complaint to BA but leave out a lot of what you wrote. Complain about the Wheelchairs and the bumping down to Economy as those are the real problems.