The Heathrow Southern proposal was for 2tph from each of Basingstoke and Guildford. (Quite where those trains would terminate at Basingstoke and Guildford was unclear, and could be the subject of another thread titled ‘projects that haven’t fully thought through their service plans’).
So, if you are travelling to Heathrow from beyond either of Portsmouth and Basingstoke, you have to change trains into a half hourly service there, or 4tph service at Woking. Interchange penalty. Then from Woking to Heathrow was not going to be quick - fitting in with the existing slow line service, thence via Chertsey. Then, dwell time at Heathrow 2-3 minutes at each of the stations, before getting on to the GWML. (Aligning paths from the SWML to the GWML is a non trivial issue either, expect pathing time a plenty). Then the journey time to OOC, which would of course depend on whether the Heathrow Southern trains were extensions of Crossrail or HEx. Then the interchange at OOC.
It all adds up, and you have to change twice, one of them into a 4tph service at best. In comparison, staying on the train you are already on, changing onto the Northern line at Waterloo (20tph minimum) and boarding at Euston - whilst longer in mileage and is also two changes - can actually be quicker in generalised journey time.
Of course the situation is different for passengers from Basingstoke, Guildford and Woking, and the intermediate stations the Heathrow service was proposed to call at (which was not many).