Virgin Atlantic have been doing this for years. The contract mentions 'London basing' - and that's it! However, after pressure from the CC and intervention from the CAA, eventually what was agreed is that you would report to LHR for LGW departures and the 'Limo' would depart from LHR 1 hour before the normal check in at LGW.
The realities of this however were that the Limo was in fact the regular shuttle bus that left on the hour every hour. This meant that your real report time at LHR could be 1.55 before check in at LGW. Considering that had to park in the staff car park by the 27R threshold and get the not too frequent bus to the Queens building, this could easily add a total of 3 hours to your day. Just what you need before operating a 2 crew sector to Florida which could be as long as 11 hours block time! they got away with it by incorrectly (in most peoples opinion - except for the CAA's!) applying the 'Florida 2 variation'.
Of course, for people like myself who thought they were joining for a LHR base, the prospect of adding 3 hours to you day was just too much, so I drove round the M25 myself to LGW, a saving of 2 hours on some days- which is what the company wanted in the first place.
I was one of the reasons I left the company.