They could still fly the aircraft within the UK on a restrictive Permit to Fly, rather than on a type certificate.
That does not require manufacturer support, it's similar to US Experimental classification. But it would not permit carriage of passengers.
The only technical reason (technical not economical) Concorde doesn't fly anymore is because the type certificate, aka the airworthiness of the aircraft, was surrendered by Airbus, hence grounding all the fleet and making it unlawful to fly it.
So to fly it, they would need that to be reinstated.
For the Avro Vulcan, they went through a different aerospace company for that.
I'd love to be wrong, but I doubt Airbus would let anyone fly one of its Concorde...