If you're travelling light, the following IMHO give the best combination of robust connection and speed:
1) Heathrow Express to Paddington where you head straight for the tube, namely the
2) Bakerloo Line (southbound) to Oxford Circus, where there's cross-platform interchange with the
3) Victoria Line southbound to Victoria station to connect with
4) Gatwick Express.
Does not involve travel via Mornington Crescent (I win!).
A taxi will be faster (not by much), more expensive (not by much
HEX and GEX ARE expensive), but the above solution is immune to M25 SNAFUS, and the tube lines involved are (in my experience) less prone to delays than others.