About half of Crossrail through the core are currently proposed to be turning round at Paddington. Meanwhile the Express are starting there. These are just obviously asking to be joined together. There is however no junction from the Crossrail tunnels into the fast lines, and interleaving some of the trains from a high-frequency core with the long distance ones starting from Paddington would be a capacity sapper.
Meanwhile the western route into Heathrow is also going to be joining the relief lines at Langley, not the main lines, so Heathrow Express are unlikely to want to get involved with that, nor to extend the current Express trains.
My own feeling is that the whole thing will be wrapped together, with a go-ahead for the new runway and a go-ahead for the western rail access, being all rolled in to the ending, possibly before 2023, of the Express contractual arrangement, and merging that into Crossrail. It won't be possible to run this nonstop on the relief lines but the timesaving from not having to change at Paddington to the tube will be a big offset for many. TfL have also long been criticised for Oyster not being available on the Express, and that will doubtless be part of the deal, which I anticipate is being negotiated behind closed doors as we write here. It will be the end of a range of nonsenses, including those foreigners who have just been sold overpriced Express tickets boarding the Connect at Heathrow Central, because it does say Paddington on it.