Won't Crossrail be using the same stations that the Heathrow Connect service uses currently? That takes about 30 mins from Paddington now, but only runs every half an hour. I reckon the Connect service will go when Crossrail comes in, but there will still only be 2 trains an hour that go the Heathrow.

The whole Express to T5 and Connect to T4 thing makes a mockery speedy connections.