My experience of this with my dad was that Heathrow was not good outbound - fine getting to lounge but then wheelchair didn't show up at lounge despite the BA agent (who was very good) calling the provider repeatedly! Wasn't just us - there were others. He felt embarrassed (he needs it for distances as he has a lung condition). The end result was the wheelchair man arriving and him then needing to go elsewhere. We arrived at the gate when boarding was well underway.
Coming back from Lisbon, despite mobility assistance showing in MMB AND me having re-confirmed it at LHR special assistance check in, the check in agent said assistance had only been requested one way! Showing her MMB didn't resolve it,I then got a "well you should always phone to confirm", when I said I did speak to BA in London to double check and we could spend ages debating itbut ultimately he needs assistance so I'd be v grateful for her help, she did apologise but it meant despite arriving at Lisbon 2 hours + before, we were stuck at check in waiting for a wheelchair until check in closed 45 mins before. I kept going back up every 15 mins and the agent would call the provider each time (called "My Way") until check in closed and I could see her packing up. At this point she was on two phones and her colleague went off to the My Way office - and we did make the flight and I think basically commandeered somebody+ a chair. Asked the CSM on boarding whether we were in the system for arrival into Heathrow - looks at ipad "oh yes, we are expecting you". Arrivals service was very good.
We're going to have to rely on this service going forward and it is stressful because, based on this, I don't kmow how reliable it'll be! I wish there was a way I could borrow an airport wheelchair & push it myself as at least I'd be in control of the process.