VS did have good codesharing as well. In addition to themselves and Delta (offering pretty comprehensive transatlantic coverage), they also had Singapore Airlines, Air New Zealand, Transaero (Russia), South African Airlines and Air China at the very least. Between all of these, they covered generally the same routes that the MAN codeshares operate (VS operate to Hong Kong, so CX would be a duplicate). VS/DL will want to feed to the long haul network via direct EDI and GLA to JFK flights, and I don't know if Virgin would want to see another airline succeed where they failed so may be reluctant to support flyBe. TCX is obviously an irrelevance at LHR.
So the market would be limited based on their existing codeshares. They'd need to find more in order to make any sizeable dent into the market - Air India would be a good choice to start with perhaps? They already do that at BHX, and it's a destination not widely served in the UK. With a large Indian population across the country, there'll be plenty of people looking to visit family and friends.