OTL
Metropolitan Vickers received its first contract for a marine gas turbine in 1943.
MGB2009 was fitted with one Metrovick G.1 Gatric, but it's unclear whether trials began in 1945 or 1947. AFAIK this was based on the Company's F.2 jet engine. According to Navypaedia she now produced 5,200 shp, but it is unclear if that is the G.1 alone or in combination with the 2 Paxman diesels.
Next were the Bold class fast attack craft, Bold Pioneer and Bold Pathfinder. According to Navypaedia, both were laid down in 1950, launched in 1951 and completed in 1953. They initially they had 2 Metrovick G.2 gas turbines and two Mercedes diesels, but the latter were replaced by a pair of Delitcs. AFAIK the G.2 was also based on the Metrovick F.2 gas turbine. According to Navypaedia the G.2 produced 4,500 shp.
Back in 1946 it appears that English Electric and Rolls Royce were both awarded contracts for marine gas turbines. The resulting EL60 and RM60 seem to have been in the 6,000 shp class.
The RM60 was fitted to the steam gunboat Grey Goose and trials seem to have taken place between 1952 and 1956. Although the trials were successful the RM60 did not become an operational engine.
Meanwhile plans to fit the EL60 to the frigate Hotham were abandoned in 1952 because of problems with the prototype ashore.
2 Brave class fast attack craft were launched in 1958 and completed in 1960. They used 3 Bristol Proteus gas turbines producing 3,500 each.
Next we have the 7 Type 81 frigates laid down 1958-60 and completed 1961-64 and the 8 County class destroyers laid down 1959-66 and completed 1962-70. Both had combined steam and gas (COSAG) machinery. The gas was provided by the G.6 gas turbine producing 7,500 shp, but it was built by AEI, not Metrovick.
The Type 14 frigate Exmouth was taken in hand for gas turbine trials in 1966. She was fitted with an Olympus rated at 15,000 shp and two Proteus engines rated at 3,500 shp each. The arrangement was called Combined Gas on Gas (COGOG) because the Olympus and Proteus engines could not be run at the same time. She rejoined the fleet in 1968.
TTL
The trials of the G.1 aboard MGB2009 and the G.2 aboard the Bold class continue as OTL.
However, Metrovick receives a contract to develop a marine version of its F.9 Sapphire engine in 1946 in place of the OTL contract awarded to Rolls Royce. Trials of the Marine Sapphire take place on Grey Goose 1952-56.
In 1946 Bristol receives contracts to develop marine versions of the Proteus and BE.10 Olympus in place of the contract awarded to English Electric.
A pair of fast attack craft fitted with three Proteus engines is ordered from Vosper at the same time as the Bold class for competitive trials. These ships are effectively the Brave class brought forward about 10 years. The two prototype Proteus boats were successful enough for 18 production boats to be built instead of the OTL Dark class, which had Deltic diesels.
HMS Hotham or another suitable ship was fitted with one Olympus and two Proteus engines. Trials began in 1953. These were effectively the Exmouth trials brought forward 15 years.
By about 1955 the results of the trials being conducted by Grey Goose and Hotham were successful enough for the Admiralty to decide to build a fast frigate with COGOG machinery. Full power was to be provided by either two Olympus or Sapphire engines up-rated to about 25,000shp. The cruising engines were to be marine versions of the Bristol Orion (first run 1956) or Rolls Royce Tyne (first run 1955).
The winners were the Bristol Olympus and RR Tyne. The research and development was paid for with the money spent on the G.6 IOTL.
A grand total of 42 Type 20 frigates were built for the Royal Navy in place of the OTL Rothesay, Tribal and Leander classes. Type 20 was effectively a gas turbine powered Leander because it carried the same armament and sensors. All other things being equal 9 would be exported (2 Chile, 3 SAN and RNZN), while the Dutch and Indians would build 6 each in their own yards. However, I'm not sure if the Australians would build 2 in place of their final pair of Type 12s.
The TTL County class would still have COSAG machinery, but the quartet of G.6 gas turbines would be replaced by a pair of Olympus engines.
Note
Some of the OTL dates may not be 100% accurate.
IOTL the Australians didn't lay down their first Type 12 until 1957.
Therefore ITTL I think that there's time for the Australian Government to cancel the entire class in favour of the British Olympus-Type COGOG frigate.
However, they then decide to build an enlarged version armed with the Tatar missile fired from a 40-round Mk 13 GMLS with 2 SPG-51 radars to guide it.
This is effectively the Australian Light Destroyer (DDL) project of the 1970s brought forward to the 1960s. The main difference apart from the 1960s electronics would be an American 5in Mk 42 or British twin 4.5" Mk 6 in A position instead of the 5" Mk 45 planned IOTL.
ITTL 9 Light Destroyers were built in place of the 6 Type 12s built in Australia and the 3 Adams class bought from the Americans IOTL. Another 3 would be built in the first half of the 1970s to replace the 3 Australian Darings.
This would give the RAN a homogenous force of 12 Light Destroyers in 1975.
This would butterfly away the 4 Perry class that the RAN bought from the USA. The money spent on them IOTL might be used to buy a replacement for Melbourne or 4 additional Oberon class submarines.
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Earlier in the thread I suggest that the British build a Terrier/Standard ER armed version of the County class in the 1960s. The missiles, launcher, magazine, radars and electronics would have been built in the UK under licence.
Now I'm going to suggest that the British build a Tatar armed destroyer with Olympus Type COGOG machinery in the 1960s.
It would effectively be the long-hull version of the Type 42 armed with Tatar instead of Sea Dart, a twin 4.5" Mk 6 turret in A position and 1960s electronics. Though the single 4.5" Mk 8 could probably have been invented earlier had there been a requirement for it earlier.
50 would be built for the Royal Navy. These would consist of: 8 instead of the County class; 9 instead of the Rothesay class; 7 instead of the Tribal class; and 26 instead of the Leander class.
Thus the RN of the early 1970s would have had a "frigoyer" force of 76 ships consisting of 50 Light Destroyers and 26 frigates (6 Type 12, 12 Type 14, 4 Type 41 and 4 Type 61).