For an imaginary train between Paddington and Southampton (anything on the potential southern link wouldn’t get past Basingstoke), the revenue would go to wherever is operating it. But the issue is it would, presumably, take Heathrow Express paths, which Heathrow Express have the rights to, so they would keep that part. Then presumably, the Southampton - Paddington train would pick up revenue for intermediate stations from Southampton to Woking, and there are other operators who have that revenue now, and would have something to say about it. Also presumably, it would take some (albeit not much) SWML to central London revenue, which would also affect the operator of the SWML franchise.
Most importantly, were the Southampton - Paddington services be a non-franchised operator, like ?Heathrow Express, then whilst the money they earn could pay for the scheme, it would be at the expense of some revenue currently assumed in the franchise model for the SW franchise and XC franchise. Which means that these franchises would pay less premium to Governement. So, in effect, the Governemnt helps to pay for the southern link, even if it is supposedly ‘privately funded’.
Of course it isn’t privately funded, it’s privately financed, which is different and then Governement will ask itself why it is effectively funding major infrastructure at private finance rates, when it could do so at public rates.