You could ask at check-in at SFO to see if VS will check your bags through to ORY. If your bags are checked all the way through, you can stay airside at LHR and follow the purple Flight Connections signs from T3 to T5. You would got through security in T5, but no need for immigration/customs.

If you bags are not checked through, you will have to go through immigration and collect them in T3 and exit landside through customs. You can then take the free Heathrow Express transfer train the T5.

This all works the other way around to (ie. ORY-SFO).

Your flights on both VS and BA will be in business class (VS do not have first class, and BA do not operate first class on shorthaul flights).