I suspect it will be another "assymetric" demand to and from Heathrow, where at Heathrow itself both services are from the same platform, and Crossrail right across London will take more than coming the other way, where people may have made a deliberate decision to go to Paddington for the Express.
Will you have to commit to an operator and a ticket/touch-in before getting to the platform at Heathrow and seeing which is coming first, or how long to the other? I suspect there will be many, particularly visitors, for whom, seeing both show going to Paddington, will just get in the first train regardless.