We should state several things about Ernie King from the get go and then examine his professional actions as relates to the UK in that context.

The first is that frankly he was a plain old son of a bitch who never forgave or forgot a slight. He was also a hard drinker, liked to dance with other officers' wives, and even lived on a yacht at the Washington Navy Yard to make it easier to do both. He was a bitter, mean spirited perfectionist who had a list of enemies a mile long even in his own Navy.

Consider this quote of his from his time as a 2 star facing retirement just before the war talking about his crews and staff:

I don't care how good they are. Unless they get a kick in the ass every six weeks, they'll slack off.

Moving on to the British, King was incredibly mum on most personal matters, but so far as we can tell the animosity began to come out in WW1. He was on Admiral Mayo's staff and interacted often and routinely with his British peers. But it was also a professional matter, he simply had a VERY low opinion of the Royal Navy by the time 1941 rolled around. He didnt trust that they could live up to their commitments without massive US support, thought some of their plans for the European theater were using American blood to protect a post war surviving British Empire. While he also had seen many of their plans proven to be total house of cards, none more so than the stunningly rapid fall of Singapore in early 1942, collapsing any chance of a robust Allied defense of the Dutch East Indies.

He was also very defensive of the amount of resources sent to the Pacific vs the European theaters, while he had an opponent perhaps none more his opposite than Alan Brooke, chief of the Imperial General Staff. He also came away from the initial joint US-British command meetings that the British were schemers looking to hoodwink their new Allies into plans of their own design. Part of this is probably true, and some honestly for the best such as using North Africa to blood US forces before mainland Europe. But some was also the contrast between the nations as the modern US Joint Chiefs of Staff had yet to really form as a coherent body. But while he never doubted the validity of 'Europe First' it did not to him mean no offensive action against Japan, and that Japan would be defeated less than 6 months after Japan, the wisdom of this view of going on the offensive in both theaters has some good level of merit.

So yeah sometimes his antagonism and mistrust of the British hurt the war effort such as in response to DRUMBEAT. Elsewhere it was born out such as the abysmal going of the British in the first year in the Pacific despite all assurances. Like much of the rest of his professional life he was incredibly prickly and seemed committed to only change his mind once made up in the most dire of circumstances.