"Would the tunnel for the rail-link to the West be the same tunnel as the spur to Heathrow for HS2 ? Or is that another one ?"
Doesn't look like it - the Langley tunnel comes in from the NW whereas I suspect the HS2 spur will have to come in from Old Oak Common almost certainly along the Heathrow Express Link - they're in the process of adding another bridge over the West Coast Main Line on that routing for crossrail services and a bigger station at H&H