CSAR, such as it is, has been the preserve of RAF and RN SH since about 2005/6. It was never the preserve of RAF SARF.
The RN will still require SAR at sea as they have always done and, until CVF arrives, the capability will continue to be covered by front line RN helos at sea (viz this year's GAPAN trophy). The shore based RN force has always been on the back of the sea requirement, unlike the RAF SARF, and is why it comes as little overhead to the RN.
So long as CVF gets a single engined jet it is quite likely that there will be a SAR reqt on CVF....but this could be done by contract (as could the VERTREP requirement).
A very sad day for both forces but the writing has been on the wall for years.