Hell, some Scandinavians (and other nationalities, especially Dutch, IME) speak better English than people born here in the US. 
But more to the point, actually, I do know a number of singers who barely speak English, but can sing it rather convincingly. It’s kind of odd, but it doesn’t seem to be all that uncommon.
I worked with people from Germany, Russia, Ireland and Colombia. None of them had any accent at all. All sounded like any other American. The woman from Ireland said she got rid of her accent as a kid because people made fun of her.
Fingerstyle guitar genius Gabriella Quevedo. Sweden.
I find that surprising and a bit atypical, at least in my experience, unless they came to the US at a relatively early age. But I also love picking out accents, so I might be more sensitive to it. (I grew up in an immigrant neighborhood, and I lived for 6 years abroad, so I’ve been exposed to all sorts of English. I could think of maybe three or four people I’ve met out of thousands in that time where I would say they absolutely had zero detectable “non-English of some sort” accent to me.) With ABBA, I can definitely hear their accents especially in songs like “Money, Money, Money,” but singing does seem to minimize it in some way.
After hearing Long Cool Woman, I thought The Hollies were American. Nope, British.
One thing is many regional and foreign accents in English are distinguished to a greater or lesser extent by intonation and prosody. When singing, these features are often forced by the song, so the accent of the singer is less clear.
They all came to the US around age 18. The Russian knew no English when she got here. The Irish woman just needed to change her accent to American since Ireland is English speaking . We also had a local NHL Russian player who had no accent don’t know when he moved here.
On the other hand I knew a nice guy from China and I could only understand about 10% of what he said He and another Chinese woman were told to get better at English or lose their jobs. They took English classes which helped a bit.
Using the Wayback Machine…
A lot of people (including me) initially expected The Buckinghams and The Royal Guardsmen to be British.
I initially thought Green Day was British. Billy Joe Armstrong often does sing with an accent.
Faux Punk accent.
How about the Beau Brummels? IIRC they were a San Francisco folk group who electified, adopted a British Invasion appropriate name, just in time to see the “San Francisco sound” become the hot phenomenon.
Thought for many years that The Cult were American-Southwest, to be exact. A combination of their former name [Southern Death Cult] combined with their fascination with Native American mysticism. Doesn’t help that lead singer Ian Asbury talks to American audiences in between songs with an American accent…
My entry also goes way back. “Venus” (“Well, I’m your Venus, I’m your fire and your desire”). Shocking Blue were obviously singing phonectically, and I somehow got the impression that they were Swedish. Ten years later I learned they were Dutch.
Why obviously singing phonetically? Lots, probably most, of the Dutch speak perfectly fine English.
I’m pretty sure I only learned about 10 years ago that Al Stewart was Scottish.
Funny. They were a local bar band around here with a pretty good following before they took off for England and got famous. They went by the name of the Tomcats. They’re all from Long Island.
I guess I was thinking Stray Cats were British because that is where they became well known. and they looked British too. 
Would that have applied in 1967? (Genuinely asking.)
Here’s my flimsy evidence:
a) To me the record sounds like someone singing a language they don’t understand. It doesn’t sound like perfectly fine English with a Dutch accent. Coincidentally, English spoken with a heavy Dutch accent is something I would recognize, for reasons not worth going into.
b) Back in the day “I read somewhere” about the singing being phonetic. But for all I know that also might be where I got the notion the band was Swedish.
So if it turns out I’m wrong, no biggie.
Full Disclosure: that song is one of my guilty pleasures. I still love it half a century later. Makes Mrs. Cretin cringe when I listen to it.
He grew up in the north-east of England and so for me, he has a clear “geordie” twang, though obviously softened by his 40+ years of international touring his accent remains firmly “nyercassle”.
Indeed, the music for Newcastle United as the teams emerge is his “going home” from local hero.
Let me give a couple of examples from acting (which I know better) of people who are known as being one thing but who spent the first several years of their lives elsewhere. Saoirse Ronan was born in and spent the first three years of her life in New York, although her parents are Irish citizens. The family moved back to Dublin after those three years. Nicole Kidman was born in Honolulu and lived there for a few months. Then she lived in the Washington D.C. area for the rest of the first four years of her life. Her parents are Australian citizens. Her father was finishing his Ph.D. at the University of Hawaii when she was born. He did a post-doc at the National Institute of Health in Bethesda, Maryland. The family then moved back to Sydney.