UK bands have been very successful in breaking the American market: the top 4 best-selling bands of all-time are from the UK and the key to that success has always been the easy transition to the huge American market. However there have been a few UK bands that have enjoyed notable success and acclaim in the UK, but are virtually unknown in the USA or have a much-reduced profile.
My top three (and this is more about personal preference than using an objective measure):
The Stone Roses - hugely influential Manchester indie band of the late 80s/early 90s who took indie music in new directions. Minor indie success in the USA, but widely regarded as the most important band of their time in the UK
The Jam - Surrey punk/new wave band. Regarded as having a similar stature to the Clash and Sex Pistols in the UK, very commercially successful in the UK and regarded as the key force in the evolution of new wave. Barely made a dent stateside.
Adam and the Ants - London based new wave/pop band. Originating from the early days of UK punk, they became an all-pervading force in UK pop music in the early 80s, dominating the UK charts. Stateside they were not unknown, but their biggest UK single and classic hit, Prince Charming, failed to even chart.
Not a particular favourite of mine, but nevertheless I was surprised to find that Madness - one of the biggest singles acts of the 80s over here - were a one-hit wonder in the US.
I’ll nominate The Move. A popular band from Birmingham, with seven top-ten hits in the UK in the late 1960s and early 1970s, but they never made the top 40 in the US.
If the band is known for anything in the US now, it’s that it was the forerunner of Electric Light Orchestra.
Not a band, but Cliff Richard is a good example of massive popularity in the UK with just minor success in the US. He did have two top ten singles in America, but according to his wiki, he is third all-time in UK singles sales behind only Elvis and The Beatles.
In the US, they’re mostly known for Run Runaway, which came about relatively late in their career (but made it to #20 here). At about that same time, Quiet Riot made it to #5 with a cover of a Slade song (Cum On Feel the Noize), having more success here than Slade ever did.
They never had any hits in the US, but their albums sold well enough, especially after their performance in Woodstock, which was a highlight of the film.
My favorite, the Bonzo Dog Band, never broke in the US.
The entire Canterbury scene of groups were too esoteric to succeed in the US, but were very well regarded in the UK: Soft Machine, Gong, Caravan, Hatfield and the North, etc.
How about Marillion? “Kayliegh” was a minor radio hit here in the US, which introduced me to the band, and I’ve been a fan ever since. My understanding is that they were absolutely huge in Europe. They’re virtually unknown here.