This very question appeared in this months Business Traveller rag.

Short answer: it may be feasible in theory, by using the existing Brighton to Reading line, new dual voltage trains would need to be acquired or the electrification of the Great Western section as far as the HEX spur. Diesel electric trains are banned from Heathrow's subterranean station.

Even shorter answer is: nah, itsjustnotgonnahappen