The challenge with code sharing between airlines and railways is the interchange between rail and air.
Virgin Atlantic's main airport Heathrow isn't really rail connected (other than reversing at Paddington or via cross rail).
The HS1 route also does not connect to any major airports other than Paris CDG in France.
Whereas if you were looking to codeshare actually Birmingham international would be the best interchange from virgin plane to virgin train. Otherwise a code share should include a connecting ticket on a partner railway (like Renfe in Spain).
Whilst it's not a current plan, the Western Approach to Heathrow would make direct connections to quite a chunk of the country possible (either direct or with a change at Reading).
Even utilising services on the (also yet to be implemented) Southern Approach, whilst it wouldn't really allow code sharing, you could create a Virgin train service.
This could give Virgin Atlantic passengers a "code sharing" service from Basingstoke, Woking and Guildford. There's even the possibility of them getting on board perks, this could include coffee and a snack, reserved coach (which might only be closest to the exit rather than a typical first class offering), luggage check-in (on the train would allow people from any station to use it and wouldn't require a dedicated space at a station, however at station could be seen as a more premium service).
There would be quite a few people who would pay (say) £20 for their "flight" to start at a station much closer to them, not least as it would save then far more than that in taxi costs. That's before you consider the removal of the stress of traffic delays impacting your ability to get to your flight (assuming that once at a station before a certain departure time that then means you have the same protections as it's you were undertaking a connecting flight).