All the options explored in the last 40-odd years have been nixed by everyone. Govts, business, airlines, locals. No one could agree. STN expansion did not happen, LTN is hemmed in and across two counties, one of which is against expansion. LGW story is well known. The Thames Estury was NEVER going to be viable.
One of the best options, I think, was to repurpose Greenham Common airbase, just to the SE of Newbury. Then put a fast (NOT 'high speed'!) rail link up to the Paddington and down to the Waterloo lines, Didcot and Basingstoke respectively. There might have been an Express on it's own line to Paddington joining in at Reading. The A34 was already going Dual and could link M3 + 4 and A303, as it does now.
As I recall, when the airbase was due to be handed back by the USA, the PM of the day (Thatcher) responded that they were obliged by ancient law to return the property as public land. The RAF base closed in 1992 (Wikipedia). Old laws can (and often are) changed. But I think that, apart from local protest from those who can afford to live there, the politicians were probably anxious about more protests, given the recent history of public protest there.
As to LHR ... one thing I like about T3 is that - whilst it might be a dump - I know my way around! I first went through the Oceanic terminal with my family in 1965. Likewise for T1, first there Dec 75, I think. I have only used T4 twice [once for Concorde
]. T5 is OK and I know my way around. The new T2 is very good.
We all know that plans have been drawn up for the rest of T2 advancing on to the old T1 site - to give space to replace T3 and turn the whole central area into a 'toast rack' to maximise utilisation. Given money questions, I'm guessing it will be a while.
We all know that there has never been a consistent approach to public planning and transport (HS2) but we are where we are and no one should expect any change. Just to repeat - LHR R3 will NEVER be built.