I make the total passenger numbers for the airports to something like ~130m/yr.
Assuming passenger numbers counts a round trip twice (I can't get any information on that topic?) that is 65m passengers per year in each direction.
If the airport is open to 360 days a year that is roughly 180,500 passengers per day or ~9000 passengers per hour.
If we assume a 50% capacity factor but with only 50% of passengers coming out of central london that leaves us at 9000 seats per hour.
400m double deck trainsets could easily get ~1500 seats with such short journeys as the ones we are proposing, which means 6 trains per hour or something.
So something like 3 platforms with rapid turnarounds?
A better idea might be an attempt to work out how many people use the rail services to the existing airports, and then assume that the remaining passengers either use the parks and rides or drive all the way to the airport.
EDIT:
Gatwick Airport station has 13 million passengers per year
Stansted Airport station is 3.8 million passengers per year
Luton Airport station is 2.4 million passengers per year
I have no information that is reliable for Heathrow station.
So that is 19.2 million passengers plus however many for Heathrow.
I will just fudge Heathrow by adjusting the 68 million passengers to about 44 million (to account for the transfer passengers), then set the number of passengers using rail to about 40%, taking us to 17.6 million passengers for Heathrow.
Giving us 36.8 million passengers for rail services out of the centre of the city. (Ofcourse some Gatwick passengers would be coming from the south).
That takes us about a 100,000 passengers per day, or roughly 50,000 passengers each way, or roughly 40 full trainloads a day, or perhaps 4 per hour.
EDIT #2:
I say we could set:
6tph Central London
6tph Gatwick
6tph Heathrow
6tph Luton/Stansted (stopping twice)
Which leaves plenty of room on the loop for other high speed services and international runs. As the lines out of the airport have 36tph potential capacity and majority of the loops would be loaded at far lower than the 24tph that suggests on the face of it.
You are out by a factor of two on most of those numbere HSTEd.
A 'London Hub' airport would need to handle at least 150m passengers per year, projected to more the 200m by 2050.
The Foster aiport proposal for the Isle of Grain - which would have involved transfer of all the scheduled business from Heathrow and Gatwick- incoroporated a HSR station to handle 300,000 passenger movements per day, plus a Crossrail link (onto the South East arm) to handle 150,000 passengers per day. Foster proposed 12 HSR platforms (on two levels) and two Crossrail platforms. It would be the largest mainlline station in the UK by passenger volume; slightly larger than Waterloo.
Granted that the Foster HSR line would also have served his orbital freight line; his passenger volumes are about right for a single London Hub located to the east of the city. He was assuming that a bit less than half of the passengers using the station would take an HSR link into central London (quite where was not clear, but probaly St Pancras); and the rest would arive on a semi-orbital HSR line alongside the northern half of M25 as far as Heathrow. Passengers from outer London, and the rest of the UK, would take the M25 to one of a series of entry stations along the HSR line (where all the long-stay parking and car hire would be located), and would then take a train into the hub airport.
The point is that, if the airport is on a island or on reclaimed estuary land, there can be no parking on site. The nearest parking would probably be across the Thames at Tilbury. Everyone (400,000 to 500,000 passengers pe day, plus 100,000 employess) would need to take public transport or a taxi to get onto the the airport site itself.
Numbers would be considerabley more manageable, if Stansted were to be chosen as a hub airport site; on the assumption that this would be a 'second' hub with Heathrow remaining open (as at Tokyo Narita). Heathrow has the capacity to handle over 80m passengers per year (the current impression of its being full is an artefact of the airlines deliberately using smaller than optimal aircraft, in order to fill their slots). Hence a Stansted hub would need to be able to accommodate 70m-100m passengers per year at most. Moreover, it is likely that a higher proportion would be able to drive and park near the airport; and (being on the north side of the city) a proprotion of trains could run through to Cambridge, Peterbough and the ECML.
A Gatwick hub would be a bit more problematic; mainly because there is propably not enough space to expand the airport site itself to full 'hub' status. But it could certainly handle 50m passengers per year with an extra runway - which might be just about enough. The problem for Gatwick is that it sits across the Brighton line, which is chock full as it is; and fitting a second HSR line into the City from Gatwick would be much more expensive than constructing one in from Stansted.