I thought it was the NATS not the CAA that is closing the air space?
It's neither.
Only the government has the authority to close airspace. In fact technically airspace isn't closed, but the CAA (as the regulator) has told NATS (as the ANS provider) that it is not authorised to offer an ATS service in the affected areas, which effectively means no IFR traffic and therefore no airline operations.
GA aircraft are still free to fly, outside of controlled airspace, right through the ash cloud if they so desire ...