Yes tunnelling is very very expensive, especially if critical facilities are on top of it!!
The LGW plans require no public access tunnels just a few minor under tax way tunnels for ramp traffic, little more than a culvert with a top on. The A23 which currently goes under the airport to the west of the rail link will be moved in an arc back towards the M23 with all long stay car parks served from it and a new airport link to the M23 at the current site of J9.
From the plans I have seen I very much doubt they will be talking about a bored tunnel using TBM's The ground around the north and west of the airport is actually pretty vacant despite what people would have you believe. The tunnelling works outside the airport would involve diverting a section of the A4 underground and then creating a covered section of the M25. Both of these would likely use the cut and cover method. within the airport there would probably be at least 6 bored tunnels using TBMs but these would be paid for privately.
Lets not also forget that Ferrovial is also one of the worlds leading tunnelling contractors and just built the tunnels for a major part of Crossrail amongst others.