Except it won't. Any money spent on Heathrow isn't taken from the regions as that amount of spend would never be envisaged for the regions. The UK is and always will be London centric.
You do realise the areas set to benefit from this tunnel are all out to the west? It isn't really of any benefit to London is it. People travel to LHR from all over the South West and many come down from Brimingham into Reading. These people are all inconvenienced by having to go to Hayes and change before travelling back the other direction and down to LHR. Calling the Western Access tunnel a 'London Centric' project doesn't really make sense.
It is also something that has been on the cards for a very long time so hasn't just jumped to the front of the queue because it happens to be in the SE.