I sometimes hear Marc Bolan and T. Rex mentioned as a one hit wonder in the States, and Bolan started in the 60s as half of Tolkienesque folk-duo Tyrannosaurus Rex {with Steve Peregrine Took, no less} before he invented glam and took off. T.Rex had four number one singles and eleven songs in the top ten, and “T.Rexstasy” was a legitimate cultural phenomenon. Bolan’s influence and impact were vast, T.Rex are still remembered with huge fondness, and yet in America he’s that guy who sang Get It On.
Lola, All Day and All of the Night, and You Really Got Me are still in continuous rotation on classic rock stations.
Roy Harper
He’s technically heard in the U.S. on “Have a Cigar”.
Those, too. Weird Al (Happy birthday, Al!) did a parody of Lola called Yoda.
The Small Faces.
Stevie Winwood in his work with The Spencer Davis Group and Traffic
Yep, as said, the somewhat tragic Peter Green.
The glorious Sandy Denny, she worked a lot with Richard Thompson/Fairport Convention: - YouTube
I think I had a thread awhile back like “famous artists in their own right whose most famous work was with a different group” and these would certainly fit (along with Elizabeth Fraser whose most recognizable song is with Massive Attack’s Teardrop.)
The Move were a strange blend. I just heard Fire brigade today. Great stuff. They were just that smidge too english and I never felt they fulfilled the promise until Shazam, and Do Ya. By then it was too late.
PJ Proby is another big name in England who is only known for that fact in the US.
The Move was going to be my addition to this thread, as well. Possibly not the “best” band for the OP’s purposes, but one that I, as an American, have at least heard of – though the only reason I know about them is due to the ELO connection.
They had seven top 10 singles in the UK between 1966 and 1972, but their only song which charted in the U.S. was “Do Ya,” from their final album, which hit #97 in 1972. (The song was re-recorded by Jeff Lynne with ELO in '76, and became a top 40 hit for them.)
Lynne joined The Move fairly late in their history, coming over from a different band (The Idle Race).
Roy Wood was inducted into the Rock and Roll Hall of Fame with ELO this past spring; I imagine that a lot of people had no idea who he was (since he was only in ELO for their first album).
How about the Stone Roses? My understanding is they were big in the UK in the eighties and nineties. But they were and are pretty much unknown in America.
eta: Just noticed the OP specified bands from the sixties. So never mind.
Renaissance
Traffic
And cribbing from people above:
The Move
The Small Faces
The Kinks
Status Quo
Slade - just picked up a Slade Alive record for the collection.
While we are here, I think Pavlovs Dog is relatively unknown in the US despite being American. They were big in Europe. How do I know? I have a young 40yr friend from Lichtenstein (we both live in New Zealand) who knows I like music. One day he said to me - Pavlovs Dog. Dang. Most people have never heard of them but this Euro guy knows Yes and Focus etc etc. It is heart warming.
Again, the Kinks were popular in the US, had lots of airplay.
Re: Slade. Certainly, they were never nearly as big in the US as they were in England, but they did have two Top 40 hits in 1983 (“My Oh My,” #37, and “Run Runaway,” #20).
That said, it’s ironic that Quiet Riot had an even bigger hit than that by covering a Slade song in that very same year (“Cum on Feel the Noize,” #5).
No, Traffic was quite well known in the USA too. Low Spark of High Heeled Boys & Dear Mr. Fantasy are still in rotation on classic rock stations today.
There’s a huge difference between “never heard in the US”, and “weren’t as big in the US as they were in Britain”.
What is interesting about this from the UK side of the pond is that, by this point, Slade were essentially finished and certainly nowhere near their heyday. Both those songs were hits in the UK but, at their zenith, Slade were the first act to have singles go straight into the UK chart at #1 since The Beatles. They were that big.
Quiet Riot introduced America to Slade,a nd probably allowed them to have their first success here. Not that it lasted long.
They were extremely popular with the college crowd circa 1990.
This is exactly why I was suggesting Slade. I’ll admit that they barely accomplished anything in the 60s, but they were formed in the 60s and were most popular in the UK in the early to mid 70s, so I’ll grant them “60s group” status.
They were (virtually) unknown in the US until the late 70s and early 80s. I had a Slade album (US release) in the early 70s and would bring it to parties. Nobody had ever heard of them. (“And why the hell are they so bad at spelling?”)
I’ve got to stick with Slade.
I’ll agree with Slade. I’d heard of them, but they might have only had one song that was known in the US, Mama, Weer All Crazee Now, and that would be because of the peculiar spelling. They did become more well-known in the 80s because of Quiet Riot.
Hawkwind is only marginally of the 60s, but they had 20+ albums chart in the UK, and a few hit(ish) singles.
Plus they produced Lemmy.