A victorious Axis is likely to have a much larger Navy than the Soviet Union at the start of the Cold War in this scenario, and significantly more advanced in terms of actually deployed equipment. NATO in the late 1940s and early 1950s was incredibly worried about hundreds of Soviet Long-Range submarines based on the Type XXIs. A victorious Germany (judging by what I remember about
@varyar's timeline I believe this happened as a result of Britain during for peace in 1940) is likely to to to emerge from the war with a growing fleet of conventional submarines, without snorkels or streamlining, but on the other hand their economy will be considerably less damaged than the 1950s Soviet Union so they can build considerably larger numbers of them. Existing Destroyer Escort and Frigate designs and Escort Carrier Airwings can deal with these fairly easily once all the right technologies and tactics are in place.
However, something like a Type XXI (or worse a Type XVIII HTP-powered submarine) will appear, although in peacetime (and without wartime experience of Allied ASW tactics) this will probably happen slower, but the submarines are unlikely to anywhere near as crippled by the contingencies of wartime production like the OTL Type XXIs were.
At that moment all existing ASW escorts will effectively become obsolete, as they will no longer be able to catch up with their quarry, and in the case of Escort Carrier Airwings, will not be able to detect submerged submarines. This is going to have the same effect on ASW-specialised warships as it did in OTL, but as per OTL, nothing like the USS
Norfolk (DL-1) is going to able to be built in numbers, so we will be looking at conversions of wartime destroyer designs like the DDKs, Type 15s, 16s and 18s. Escort Carriers will eventually be replaced by wartime Fleet Carriers, as the former were not large enough to carry single-package ASW aircraft like the S-2 Tracker (which are the only aircraft capable of carrying the necessary equipment to detect submerged submarines).
It should be noted that one of the rationales for the USS United States (CVA-58) was Anti-Submarine Warfare, to be achieved through "Attack-via-Source" which would involve attacks on their home-ports with Nuclear Bombs or even 12,000lb earthquake bombs (both CVA-58 and the earlier 1945 Fleet Carrier design were required to have bomb-lifts capable of lifting 12,000lbs of ordnance). I expect Attack-via-Source would also be achieved through offensive-minelaying by aircraft (both land and carrier-based) and by submarine.
Surface raiders will be a significant threat in North Atlantic weather conditions until the late 1950s, where a combination of miniaturised nuclear warheads, Nuclear-Powered Submarines and All-Weather Strike-Aircraft like the Buccaneer and A-6 Intruder enter service. Given this, I expect significantly more investment in surface-ship anti-ship torpedoes like the Mk 17 and Barmaid (a proposed surface-ship equivalent of the Mk 12 Fancy) with the
Mitschers (or their OTL equivalent) looking much like their earlier design studies, with centerline quintuple torpedo tubes and larger numbers of 5"/54 guns. The high seakeeping-speeds required either to chase down enemy surface combatants or to screen the Aircraft Carriers will drive up surface combatant size as per OTL, even before the volume requirements of early missiles and their radars (Project Bumblebee was in part started to counter German Anti-Ship Missiles, so the Talos, Terrier and Tartar family are also likely to be developed, and something like Typhon or Aegis will follow when the limitations of the Three-Ts become apparent).
Gun-armed Cruisers and Battleships will probably still be built at least into the late 1940s, most likely to wards the end with auto-loading guns, and their successors will be missile armed (the large volume requirements for early missiles will require large hulls, whether those will be conversions of existing ships as per OTL, or clean-sheet designs I don't know).
The Italians are going to be a fairly significant issue. If I remember correctly, in
@varyar timeline they control North Africa, and so probably also have control of East Africa. OTL pre-war they had fairly ambitious plans to base raiding fleets of Aircraft Carriers, Fast Battleships, long-range cruisers, oceanic scouts (either large destroyers or small cruisers), fast tankers and long-range submarines. This was massively overambitious even pre-war, but Post-War I expect them to build at least a portion of that, especially the cruisers, oceanic scouts (some of which were built OTL), tankers and submarines (again some built OTL). This will require significant forces in the Indian Ocean to counter, (furnished either by Britain, the US, and/or an Allied-aligned independent India).
For the 1940s and 50s, a 600 ship navy is probably slightly too small, (Arleigh Burke in 1957 wanted the US Navy to consist of over 900 ships by 1970, but this would not be achieved OTL or in TTL given how expensive Missile-Amred ships become, combined with a Polaris-equivalent competing for funding), but is very likely to be a reality by the 1980s. OTL the US Navy shrank fairly significantly in the 1970s due to spending a decade without the construction of large surface combatants (due to a combination of Polaris, Vietnam and the failure of Typhon) and the withdrawal from service of conversions and life-extended Second World War ships (this bit would happen TTL), but a 600-ship Navy would probably already be a reality in world with a surviving Nazi Germany.