Ok! First let me ping another user who is a treasure trove when it comes to the Senior Service. /u/fourthmaninaboat

But in general we can say no. Post war budget cuts and the breakup of the Empire, along with an evolving security situation meant that the Royal Navy was already by the start of the Cold War no longer top dog. She had surrendered that title to the Yanks. The US Navy had undergone an incredible transformation in 60 years From a bad joke in the 1880's, to a credible world power, to being given essentially strategic parity to Britain under the Treaty System, to coming out of WW2 with the largest collection of warships ever dreamed of.

And while both services drew down severely the USN remained larger, and would continue to grow again in meeting its new global requirements. In essence filling the presence gap of global seapower and guarantor of maritime trade that Britain no longer had the means or ability to fill. While the Soviets commenced to a huge series of construction programs as well.

The Royal Navy also took on a new character as part of the integrated NATO naval structure. Its location allowed for a focus on ASW on anti submarine warfare. In particular it could serve to fight it out with Soviet subs on the G-I-UK line, the area between Greenland, Iceland, and the UK through which Soviet fast attack boats and surface task forces would need to pass to get at NATO sea lanes of communication.

This was not an immediate shift though, and while confirmed by the 1881 White Paper as their primary naval mission, it was in part an evolution of shrinking imperial commitments and eroding true ability to project power and proximity to the new threat.

We can also trace the erosion of the RN's forces by way of their carrier fleet. In the mid 60's the large fleet carrier CVA-01 design(which were still smaller than contemporary USN designs) was cut as too costly, and at a time when the RN operated just 5 carriers of WW2 or immediately post war vintage and it was decided that the RN would cease operating fixed wing carriers, meaning that all designs would rely on helicopters or VSTOL aircraft such as the emerging Harrier and the larger ships disposed of as the scheme to operate US built F-4's proved unworkable. Come the early 80's then when the Argentines invaded the Falklands the RN found itself stretched to the max to build a task force to retake the islands. The old HMS Hermes which had been a WW2 design finished in the 50s, then a helo carrier for the Royal Marines, then an ASW carrier, and had just recently been refitted to operate Harriers. You also had HMS Invincible the first of the new class of 3 small carriers meant to represent the future CV force. But ironically just before the war it had been announced she would be sold to Australia! This was canned and she went south with the Task Force.

With the end of that war Hermes was sold off to India actually(she served well there and actually just formally ended her career as INS Viraat last week!!), and the 3 Invincibles represented the core of the Royal Navy through the mid 90's and past the 20 year cut off. They were actually all decommissioned themselves in recent years, and until the new Queen Elizabeth class ships are finished the Royal Navy has no carriers.

A small but very limited force, with similar stories for surface combatants, and subs. They could punch above their weight class but were limited in funding and number. Clearly as second fiddle to the USN, and really even the Soviets. The mystic of the Royal Navy is still integral to their conceptualization as war fighters, and they hold onto and honor it dearly, but the realities of the budget and shrunken global requirements in a changing world caught up to them.