The true rival for the UK in the early years of the 20th Century, and the last decade of 19th, was the United States. The U.S. & UK both needed control of the Atlantic to defend their economic life blood. The U.S. was openly expansionist, having driven the Spanish from the New World, taken much of Spain's Empire in the Pacific and presented a nascent, but growing strategic threat to Malaya and even Singapore. The American Navy, while lacking in light forces possessed a growing number of absolute cutting edge capital ships equal to anything in the RN. It was this competition for national security needs that, in major part (along with both countries desire to counter the Tsar), brought about the Anglo/Japanese Alliance in the very early 20th Century. This afforded the UK (and Japan), at minimal cost, a nice counterbalance to the USN and any potential American aggression since it was a mutual DEFENSE arrangement. The demise of the Anglo/Japanese Treaty in 1921 was entirely the result of the UK deciding that it was far more useful to maintain close ties to the Americans (and not incidentally stop a naval arms race that the destitute post WW I UK had no hope of winning) than to antagonize Washington by supporting the Japanese against the U.S.
Calbear
I'm surprised your propagating this old myth.
America was a major economic rival of Britain and for historical and reasons of vested interests sometimes took an unfriendly stance to the UK but Britain had already committed itself resolving any dispute with America by any means other than war.
Steve