Not actually been there, but I imagine Wolverton or Northampton are the closest stations, but I can't see either being especially close.
I reckon the closest First Group served station would be Bedford, which would be something like TPE to Manchester and on to Doncaster, then First Hull Trains to Kings Cross then FCC from St Pancras to Bedford. But that leaves you some way short of the circuit.
You could get TPE down to Manchester and then book a cheap Advance to Northampton, Wolverton, Milton Keynes or even Banbury. Or you could go to Luton (see route to Bedford) then get the 99 bus over to Milton Keynes.
Or, if you prefer, there is always getting a Virgin Trains service to Wolverhampton/Birmingham and changing there for Banbury, or going by Virgin Trains all the way to Milton Keynes.
Advance tickets would be the cheapest way to take the direct routes, PRIV tickets aren't particularly cheap on the West Coast.