I think the point is that they don't run the "franchise", they have a "management contract" to caretake the line until the issues surrounding the franchising process are resolved. I'm sure there'll be some legalese that allows this to happen.
From a legal point of view, thats right on the nose. Virgin wouldn't hold the franchise, but rather their expertise and management would be contracted to the DfT, with the DfT calling the shots (even as far as telling Virgin how big their logo can be on the train if they so wish). For it to work like that, all revenue from fares would be collected by Virgin and paid directly to the DfT, with Virgin being paid a fixed fee for their services, a fee that would be agreed by both parties before such a deal is signed. The other option of course, as has been discussed, is DOR holding the franchise but contracting the operation out to Virgin until DOR are in a position to run themselves, Virgin would still run the actual day to day operation but wouldn't hold the franchise. Perfectly legal to do this as long as both parties agree to the fixed fee paid to Virgin and the contract length. Of course, a management contract could be handed to any operating company at the DfT's choice, but as Virgin already operate the franchise it would be inefficient to start discussions with other operators for this.
As has been said, in this instance NO operating company has the right to operate the services, it is down to the DfT to decide themselves, of which can legally be done without a fair competition. Naturally, First will be upset at the DfT handing a management contract to Virgin, but to be honest nothing can be done in this instance. Strictly speaking from a competition perspective, a small tender process would normally be required to decide who will run the management contract, but being so close to the franchise end now, the time is not there to run a fair competition process, therefore such management contract could be declared as an emergency contract, which does not need a competition (decision made by the DfT based on which operator has the resources ready to take over the operation, which in this case is Virgin) and would normally run for a maximum 12 months, though there are exceptions. BUT until the DfT make their announcement, no one can know for sure if Virgin will even be taking over on a management contract basis, until then it is all assumptions, for all we know the DfT could have a different operating company lined up to take a management contract. The immediate concern is not First vs. Virgin, but ensuring trains continue to operate smoothly with minimal disruption to passengers.
No doubt it has already been covered, there's too many pages to try find it, but First never technically won the franchise from a legal perspective, they were only declared as preferred bidder, a nice little "cover your backside" way from the DfT, and is quite common in public service contracts, be it tendered bus services, IT service provider or maintenance provider for councils etc.
Due to this, and my own personal opinion of course, First wouldn't have much to go on to take to the courts in the event of Virgin still running the mainline from 9th December, they would only have became the franchise winner once the contract was signed. If any legal action was to be successful, it would be based on share value, but again that would most likely only be considered if the share value now, after the process was cancelled, is considerably lower than it was before First were announced as preferred bidder. If it is only slightly lower they could struggle with a case, as share value variates on a daily basis, and could be affected by their recent failings with their UK Greyhound services or bus disposals, and would therefore be difficult to prove such a small drop in share value as a direct result of the franchise process being cancelled.
I'm not against First, nor am I against Virgin, I am unbiased, but my own personal opinion is the threat of legal action in the press from First is a poor copy of how Branson uses the press to put across his point. They're clearly not happy about the system being flawed so are attempting to throw their toys out of the pram Branson style to get what they want. Virgin shouldn't have pushed everything out in the press to start with, they should have pressed ahead for the judicial review quietly and in a controlled manner, but any adult (without bias against either First or Virgin) would admit that First trying the same is just childish. Of course, those who favour First would think it was childish of Branson to use the press to his advantage, but think it is perfectly fine for First to do so.