Well LHR could never promise it, not being an airline. I am sure however that a new runway open for SH ops with an easy connection on LH destinations would create a market that would be quickly filled. Not only from the UK regions but also Europe. We would also likely see new routes to destinations that were not previously viable even for Heathrow.
I have yet to hear a decent argument against it other than spurious environmental claims, protests from the Manchester Airport fan club and those worried about house prices. All I hear is Heathrow is currently crap so we shouldn't improve it.
What is actually happening is we are allowing successive governments to get away with not making a decision for purely party political reasons and for a project this important they should no be allowed to get away with it.