The British of course recognised that a new player was in the game, specially after the US Civil War. However they never had the time to 'deal with it', even if they could have 'dealt with it'. They always had other things to do, and could not commit them selfs to get drawn into a conflict with the US.

They had to deal with France-Russia relations and later German power. They were content with giving the US extended regional power status and recognise their interest in South and Central America.

Since the Americans at this time did not try to challenge the Royal Navy and showed little interest in the main areas of the British Empire. So there was no reason to fear for naval hegemony. The US was powerful state, but not centralised and organised in a way to be a major world power player.

Now lets talk about WW1. Its a difficult subject and I want to be short. Essentially their was a power play involved between British Blockade and American interest in trade. Because the British did not just attack neutral shipping the way the germans did, relation were strained but could be mended. Wilson however was not pro english for all that, he would have liked to stay neutral. For a potentially powerful player, neutrality is a position of force, for a weak nation its on of weakness. However the germans could not be ignored in 1917.

The US expanded rapidly in troupes but also in its Navy. This was aimed against the British as well, it gave the US a position of power that the Army alone could not have given them.

After WW1 there was quite a discussion about the US. What to do about this new power who had the means to challenge or even completely outstrip the British. There were people in Britain arguing for consolidation of the Empire and mobilisation of the Empire against this new powerful competition, potentially with Japan as a pacific power.

However cooler (and smarter) people point out that there was little to be gained by all this. That the US did not pose a danger to British, different from the Germans. Not to mention that they already were in huge debt. Canada would not support such a moves either.

The US played with the new found power by calling for the 'Washington Naval Conference' in 1922. In this conference all the major powers agree to a treaty. Battleships would be more restricted then Nukes today. In this conference it was agree that the US and the British are the primary power in the world. The agree on parity in battle ships, with Japan ranking third and France and Italy last. The ratio was 5:5:3:1.7:1.7.

So at this point everybody in the world saw that the British were no longer the primary power. They had allowed the US parity in there most important area of interest.

Many people in Britain realised that the US made a much better friend then an enemy. Instead of trying to force or diplomacy they accepted a conferrable second place.

All of this did not mean that the US actually had more naval power, rather that Britain would except it, and by that excepting, they had already dealt with the reality.

Only in WW2 the US really started to far outstripped the British in actual naval power. After WW2 the British had fought 2 World Wars from start to finish, their population was tiered and wanted social reform and not a post war power state.

This is essentially the much simplified story of how the US rapidly caught up to the British in Power (as german had done) and then started to race ahead. The US has had this position now for essentially 100 year about. As the British had between Napoleon and WW1.

tl:dr; The British were forced to deal with reality after WW1 and did not like it, but were smart enough to see that it was not worth doing anything about it.

Mark Mozower in his book "No Enchanted Palace" argues a similar point to the question that your asking. His book focuses on the ideological foundation of the United Nations and, at its root, argues that Britain not only understood but accepted the American imperial dominance after World War II. Though not specifically naval dominance, imperialism tends to arise from a strong naval power.

From a logical view it makes sense. Britain and the rest of Europe forfeited many colonies after WW2 and British Imperialism would decline significantly from there while America would increase its imperial power because of the Cold War.

With this in mind, it seems that Britain was completely fine with America growing as a world power at the time because of there cultural similarities and cooperation within the U.N. There was very little competition between Western European States (I will include America in this group for simplicity) after World War II, at least compared to the years beforehand.

I wish I could answer your question about naval power directly but hopefully this answer helps.

In Lawrence James' Rise and Fall of the British Empire, the British had started to feel the financial crunch of Empire by the early part of the 20th Century. They had indebted themselves during WWI to the point only last year did they close their books on their WWI loans and bonds.

The cost of operating a modern navy grew exponentially for the English, and the cost and economic impact of the war actually sent England into a massive recession. As such England had to scale back operations and chose to reduce them mostly in the Far East theater.

Come WWII, Britain had all of two capital ships and four destroyers for it's Force Z, which was to reinforce their holdings in Singapore. They recognized that they would have to rely upon American reinforcements to secure their holdings in the region, but as we know Pearl Harbor squashed those plans.

So even by 1941, the British had recognized that could not defend the whole of their Empire by sea, informally handing over protection of the Far East primarily to the United States.