A quick rundown on missile systems (most of those were already mentioned before, but I thought it'd be convenient to make a summary).
One of the main differences the side switch will affect is an introduction of a heavy naval surface-to-surface missiles into the British inventory, and accordingly, into the British doctrine.
As I mentioned somewhere earlier, the switch falls on the final design phase of the County class destroyer, so the Counties ITTL would be different, and would likely be the first RN ships to carry the P-5 (SS-N-3 Shaddock) in this or that configuration. That would give them strike range of 350-400 km, and would form the base for combined carrier aircraft and missile strike tactics against US carrier groups.
Possibly, the same system may be temporarily
deployed on carriers in the 60's.
This line of development eventually produces the SS-N-12.
Sea Slug, is cancelled (partly because of the room that would be needed on the Counties for the strike missiles, partly because it was ridden at that moment by development problems and there was no confidence they would be eventually solved) and replaced by Soviet SA-N-1, which is relatively quickly phased out in favour of Sea Dart.
The ramjet Sea Dart would appear in many different variations, including surface-to-surface (tactically replacing exocets and harpoons), air-launched and active radar, and at the same time would develop into the scaled-up monstrous P-700 (SS-N-19) and the more compact P-800 for vertical launch cells.
For heavy air-launched missiles, the Blue Steel would likely be merged during development with Kh-22 (AS-4) and later get a smaller variant of KSR-5 (AS-6), to be carried on V-bombers and supersonic Tu-22 and the British version of Tu-160.
Since IRL the Soviets got their hands on both Sidewinder and AIM-7, the Skyflash can be developed as well, but otherwise mostly Soviet or jointly developed air-to-air missiles would be used.
The Martel/Sea Eagle series, is, apparently, based on (or at least inspired by) Soviet Kh-58 (AS-11), so there would be, possibly, additional anti-ship variants of this AS-11 in British service.
Of course, everybody, including the Soviets, copied the Harpoon and the Tomahawk, so those would be available as well, but generally I suppose we would see a much more focus on supersonic anti-ship missiles.