I think the Sea Vixen would be used in the ground attack role. Perhaps some AAM armed planes would be sent as escorts for Buccaneers and attack Sea Vixens, but they likely would be only used defensively rather than go Mig hunting.
The state of play for the Lightning force in the mid 60s was F1/F1A/F2 had Firestreaks and guns but only ~4,200l of fuel. The F3 had no guns, Red Tops but only about ~4,200l of fuel. The F3A/F6 had no guns, Red Tops but ~6,000l of fuel.
The handful of remaining Javelins in 1965-67 had guns and Firestreaks, but were hopelessly outclassed by then. Of course the RAF did have 9 sqns of Hunter FGA/FRs that might be sent and maybe tangle with NthV Migs.
I'd expect the British missiles to perform about as well as the US did, and the French R530 for that matter. However for fighters to be committed Britain would need a vastly different political environment.
I doubt the environment off NthV would stretch ADAWS very much; there would be controlling the air activity, surface search and some NGS and almost no opportunity to fire a Sea Slug.
Something that might be interesting is how RAN DDGs performed much better ELINT (I think) than identically equipped USN ships. The difference as the RAN conformed to British practice and had someone dedicated to that station whereas USN practice was to rotate people from the CIC through the station.
My guess is that different British equipment and practices would provide Task Force 77 with some niche capabilities that could be exploited when the RN carrier was on station.
