I suspect Sea Slug was ready by 1958, the reason it didn't officially enter service until 1962 is because the first ship that carried it didn't enter service until then. Most naval weapons that depend on new ships to carry them are like that, the weapons are sometimes available much earlier than their service entry date. This is particularly true of the Royal Navy (but sometimes the opposite is true with other navies, as they sometimes put ships into service without key weapon systems operational because they're not ready yet- but that's a more recent problem).
There may be something to that.
Seaslug first went to sea in the late 1950s aboard the converted maintenance ship Girdle Ness. As well as being the trials ship for Seaslug she was intended to be the prototype for a series of guided missile armed convoy escorts. Also from the late 1940s to the early 1950s the RN had plans to build Seaslug armed cruisers that would be completed in the late 1950s, but it decided to complete the Tiger, Lion and Blake instead because it was (wrongly) thought that completing half-finished ships to a new design would be quicker and cheaper.
OTOH the first pair of County class destroyers was ordered in the 1954-55 Estimates and the second pair in 1955-56. One of the reasons they weren't laid down until the late 1950s and completed 1962-63 could have been that the design of Seaslug hadn't been finalised so the DNC couldn't design a ship around it.
But that doesn't alter what RLBH wrote, which is that a joint missile might take longer to develop and what I wrote that it in turn might lead to Fighter Command having more fighters in the period 1957-62 because the guided missiles being developed to replace them weren't ready.
However, one of the reasons for developing one first generation SAM instead of 2 or 3 is to save time by concentrating the resources on a smaller number of projects. Therefore I hope that the opposite of what RLBH wrote would have happened, that is the joint missile would have entered service earlier than Bloohound, Thunderbird and Seaslug did. Therefore in my timelines where a joint missile is developed it costs as much as the combined cost of Bloodhound, Thunderbird and Seaslug, but the Mk 1 enters service in 1954 instead of 1958 and the cost savings (if any) are on the production side.
AFAIK many British military projects in the period 1945-1960 took longer to complete than equivalent American projects because they were under resourced. This was mainly because UK wasn't as rich as he USA and it didn't have the scientific and industrial resources to match either. However, the British Government compounded the problem by spreading the resources over what turned out to be too many projects.
As an example of the UKs limited scientific and industrial base in comparison to the USA, Norman Friedman in the Postwar Naval Revolution wrote that the reason why the UK didn't develop its own AEW radar in the 1950s, wasn't for lack of money, it was for lack of scientists.