The JFH harriers are all 'owned' by 1 Gp, who are RAF. They are the operating authority and have all meaningful levels of command and control over them. The RN was, as was correctly stated, supposed to joint man JFH but couldn't. Again the overall reason for this is as previously spelt out....the FAA had relatively few FJ pilots, did nothing to stem the out-flow, not much to improve the in-flow, and was left with what it started with.
At the moment the RN 'has' the Naval Strike Wing....which is basically 800 NAS augmented by some people wearing 801 NAS badges. Essentially the RN will continue to appoint people to 801, but it doesn't really exist as a unit, it's a 'budget line' for manpower planners etc.
The aspiration for a four squadron construct still exists, but whether or not it will ever happen is open to debate. To generalise massively, dark blue policy makers want four squadrons, light blue chaps want three......anyone below the rank of Wg Cdr/ Cdr doesn't care and just wants to get on with business. (Which the chaps are very good at.)
Summary, yes you can fly FJ within JFH, on a FAA squadron, but the RN neither owns nor commands the aeroplanes.
Ah! The beauty of jointery.