There's a certain logic to that you know, not from a Green perspetive but Crossrail will link Canary Wharf with LHR. It will have an impact I am sure.
Up to a point maybe, but there is one important "fly in the ointment":
Heathrow has capacity issues.
Crossrail would never be a substitute for having an airport on the doorstep.
Crossrail will certainly improve access to LHR, but perhaps not as radically as you expect. On current plans, only 4 Crossrail trains per hour will go to LHR. These won't serve T5. On top of this, likely only 2 per hour from LHR will serve C Wharf (because 50% of Crossrail services going east will go via Stratford, not the Wharf), so passengers will often need to change to get to the Wharf.
Bit stupid to have Crossrail trains running to/from Heathrow but not to/from Heathrow-5! Would that not form part of the western rail access and access to/from Reading? Surely that is not going to be left only to long distance trains?
There are many concepts of the crossrail plan that suggests an allergy to common sense, for example, why is the Greenford branch not included? why is the eastern terminus not at Southend (many commuters between Shenfield and Southend because property was cheap in that area)? why is there no station at the London City airport terminal (as the crossrail tunnel goes directly underneath)?