I think both the Western and Southern accesses to Heathrow should go ahead regardless of whether the Airport expansion does or doesn’t. The Southern access in particular will unlock even more benefits for HS2 since some trains from the South Coast can be routed into Old Oak Common via the new link. With this in place, I suspect it will become quicker to get from Bournemouth and Southampton to Birmingham and Manchester via OOC rather than using the direct Cross Country service.
Whilst I agree that Old Oak Common will be able to deliver faster journey times, via the Southern Approach to Heathrow is unlikely to do so for travel from Basingstoke or South or West*. Especially given that expected journey times are for 50 minutes from Basingstoke to Heathrow, whilst existing journey times can be as fast as 50 minutes for Basingstoke to Paddington.
* It is one of the reasons why I think that the Southern Approach to Heathrow services should be an all stations service between Basingstoke and Woking rather than a semi fast, as it's likely to attract more passengers by stopping it everywhere due to those from South or West of Basingstoke having faster journey time options to Heathrow than changing at Basingstoke. Even if that's just changing at Woking and having the option of 4tph to Heathrow.