I'd guess (and as the others have said, nothing concrete has ever been revealed, so it's just a guess), that the Joining would remain intact but become largely meaningless. The host and symbiont would still be as one, but they'd also be linked to trillions of other brains and their personality/personalities would be suppressed anyway. So they're just another component in a whole web of linked nervous systems whose resident "souls" have been locked down. Yes, they're a component with a rather odd duality - perhaps it might even register as some sort of strange integrated subsystem-within-a-subsystem? I doubt it would affect the new drone's functioning either way, and given that the Joined mind is a tiny drop in an ocean, I assume the Borg wouldn't find it particularly notable?