The VS/DL alliance includes "Reciprocal Delta Sky Club and Virgin Atlantic Clubhouse access at applicable airports for Upper Class and BusinessElite passengers, Flying Club Gold members and SkyMiles Platinum and Diamond members" according to the original press release.
I do not have experience with this, but I'd be very surprised if anyone is turned away from a SkyClub with a same-day J ticket on VS (especially with a DL flight number). I'm not quite as sure about Medallion status-based access, since DL has been reducing those benefits recently.
Please do report your experience.