When you arrive at the airport for your DL flight to JFK, give them your VS flight details/booking confirmation and then should be able to tag your bags onto the VS flight so then you won't need to collect them at JFK. Your bag receipt should should show both flights on there and final destination as LHR. It should also show in the DL app on the tracking bag section.

I would recommend doing the online check-in for the VS flight including fly ready which does the passport verification

When you arrive at JFK, just go to VS departure gate and tell them you need a boarding pass and they will want to see your passport. You can double check with them to make sure your bags your showing but as long as you have your receipt you should be fine.