When a CO codeshare exists for the flight, they will only allow you to book out of CO inventory. This is substantially less than VS inventory and is a subset of the seats that CO has bought on the flight. It's a CO reward on a CO flight number
When there is no CO codeshare, you are booking a partner reward on the partner flight number. This comes out of their inventory and I have found that this tracks pretty close to what you see on virgin-atlantic.com. Its not identical i.e. when there are limited seats <2 then VS often keeps them for their own rewards only.
We were unable to get flights on the CO codeshares last year but had to connect somewhere anyway. Since there was no CO codeshare in Chicago we had wide open availability on the VS flights from ORD-LHR.
I'm not sure if there are still any non-codeshare flights remaining. Perhaps LAS?
I flew ORD-LHR-ORD a month ago, in VS UC, as part of a SaverPass award, and while the outbound was booked with the CO codeshare number, the inbound was booked with the original VS number. (And the codeshare existed already!).
As for the people that have not been able to change their seat and other details on-line on virgin-atlantic.com, remember that their on-line system cannot deal with tickets that have more than 4 segments.