I'm new here, so bear with me, but...
I understand that running from Hayes & Harlington into Heathrow has various restrictions on what trains can be used (electric only, certain signalling kit etc), and that the replacement buses are overcrowded. For that, why not take the approach taken with other long-term disruptions, and charter in a fleet of coaches to shuttle people from Hayes to Heathrow? It will mean a change, and will be slower than a train, but ought to be sortable in under a day + relieves pressure on the buses + quicker if running non-stop
That leaves the Hayes & Harlington to Paddington overcrowding. I believe that excursion and charter trains sometimes run into Paddington. There also should be quite a few freight drivers who know the lines around Paddington from Crossrail work deliveries. So, hire in some coaches from wherever you can (eg someone who provides them for rail tours or charter trains), top an tail it with whatever freight locos you can find (diesel or electric - wires up between H&H and Paddington), then run as a shuttle into the bay platform.
Now, it won't be cheap - need to hire coaches, need to hire locos and carriages at the last minute, paperwork etc. If then-FGW did have a proper contract with Heathrow, then they pass the cost on, who likely punt it onto the 332 maintainers. If then-FGW were muppets, then GWR just has to eat the costs for their past mistakes. Or have I missed a snag with this plan?