Is there a similar short-notice procedure for such matters to allow an operator to take on work such as this, or is it a case you can only do so if you already have capacity in the business?
An operator would not be able to operate a bus for hire and reward without a valid PSV O licence disc displayed. That requirement is absolute, and there are no exceptions permitted even in exceptional circumstances such as these.
Whether it would be permissible to operate a bus without a PSV O licence and not charge fares is debatable, but an argument could be made by the TC (assuming you weren't doing it purely out of the goodness of your heart), that your "reward" would come in the future when you obtained additional licences and started charging, so I think an operator would be unwise to do this. There are also insurance implications.
That being said, most bus operators have both (a) a disc for every bus, including spares, hence can meet short notice uplifts and (b) often more discs than buses, so again they can quickly bring extra buses in to the fleet.
Not all operators in this case, but many, especially the larger ones, are.
Sadly, operators closing with no, or minimal, notice is so common across the industry, pretty much every operator of every size / scale knows what to do in this sort of situation, and is practiced at mobilising new contracts at ultra short notice. There will I am sure be some roughness round the edges as new buses, drivers and duties bed in, but compared to the previous standards of service delivery on the routes in question, I suspect passengers will be appreciative of whatever can be operated.