In London and surrounding areas most are run using TfL spec buses either still in TfL franchisee use as the weekday requirement is higher or recently retired from TfL service use so a far smaller issue than it might appear. One exception is SWR use a mix of buses and coaches if it is a big closure ( e.g. Clapham Jn - Wimbledon) but given the frequency service you would only be waiting another 2 minutes for a bus instead of a coach in this case. For smaller closures SWR are all accessible bus.
Hence I'd suggest this is mainly a not near London issue...
TfL do use non-TfL spec buses when there aren't enough appropriate spare vehicles available. In the last few months there have been appearances from Reading Buses, Centrebus, Trustybus, Redbridge Council, Arriva The Shires, Ensignbus, Go-Coach and others. The PVR on Saturdays isn't much lower than weekdays and when a lot of closures clash, other operators have to be used. Though the buses are usually PSVAR compliant.
As for TOCs in London, some stipulate minimum vehicle age which would mean all buses but those upseated for school work are PSVAR compliant by default. Others don't have any qualifiers beyond single deck/double deck, because there's often a shortage of available vehicles. When some lines in London are shut, it is very, very evident that there are nowhere near enough buses available to cover all the duties. It's not even just vehicle shortages, it's driver shortages too. Sometimes operators take on the work when they have no drivers available to do it. If rail closures weren't so erratic in nature there'd be a bigger pool of drivers who are well suited to rail replacement work. I'm a serial, dedicated (by choice) rail replacement driver and I go wherever the work is.
Many buses that do jobs in London are less than 20 years old and PSVAR compliant - but quite a few will have been upseated for school contracts and the like and are non-compliant. Another threat to rail replacement in London is looming in October 2020 when LEZ will require buses to conform to Euro 6 standards. I suspect the TOCs may have to bear the cost of the non-compliance charge as there will be nowhere near enough spare buses to comply with this. With a requirement for all RRBs to be PSVAR with a whole 2 months of notice, I can forsee some rail replacement routes being abandoned completely and everybody having to make other arrangements.