Weirdest spoken word song I can think of that was a hit was Detachable Penis by King Missile.
Though there is some actual singing in it. It wasn’t a huge hit, but it made it to 25 on Billboard’s “Modern Rock Tracks” chart.
ETA: The video above does have the “naughty bits” visually censored.
That’s my second-favorite B52s song after Rock Lobster.
ETA: I just thought of another really strange hit. The song “Oh Yeah” by Yello. It featured prominently in the film Ferris Bueller’s Day Off, and other films in the 80s. It was something of an 80s staple in fact. Countless commercials have also featured it. Despite the fact that it’s barely a song at all.
Unless you’ve lived in a cave, you’ve heard it.
That song got to #51 on Billboard’s Hot 100, and reached #8 in Australia’s Kent Music Report.
On Her Majesty’s Secret Service by The Propellerheads and David Arnold was a top 10 hit in the UK. Take the film theme, record an orchestral version (David Arnold) and then synth, sample and god-knows-what-else it to within an inch of its life (Propellerheads). I love it. Nine minutes and change - stick with it.
UK again - Gaudete by Steeleye Span. That rarest of things, a hit single in Latin:
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That singer was able to go to law school with the proceeds that song has earned over the years.
Check out the bass drum for an obscure sight gag.
Actually, I was going to post “Eres Tu,” the Mocedades hit. Like “Dominique,” it isn’t even in English but people loved it anyway.
And this one charted pretty high: Tie Me Kangaroo down, Sport.
The recording peaked at No. 1 in the Australian charts[5] and was a Top 10 hit in the UK in 1960. In 1963, Harris re-recorded the song in the UK with George Martin as producer, and this remake of the song reached No. 3 on the US Billboard Hot 100 chart and spent three weeks atop the easy listening chart in 1963.[6] “Tie Me Kangaroo Down, Sport” was a surprise hit on the US R&B chart where it went to No. 19.[7] The song reached No. 20 on the Canadian CHUM Charts.[8]
The entire song is played with one chord (C7 IIRC).
At my local grocery, in the yogurt case the coconut and key lime flavors are right next to each other. So as I’m grabbing a couple of each I can’t help but have “put de lime in de coconut” running through my head. 
I’m going to vote for Charlie Daniels’sStill in Saigon. Still In Saigon - YouTube
Also Billy Joel’s We Didn’t Start the Fire. It’s mostly just a list.Billy Joel - We Didn't Start the Fire (Official Video) - YouTube
Here’s a blast from the past:
In-A-Gadda-Da-Vida—Iron Butterfly. 1968. Hit the Top 40 at #30. The song was the entire side B: 17-minutes long and featured a 2.5-minute drum solo.
That one probably deserves an asterisk next to it, as the single version was just a 2:52 edit that cut out all of the psychedelic nonsense.
When it comes to truly long songs making the charts, the current champion is Taylor Swift’s 2021 re-recording of All Too Well - a 10:12 version of a song from an album that came out 10 years earlier and wasn’t promoted as a single at the time, which basically consists of an extremely brutal dis track directed at Jake Gyllenhaal. It made #1 on the Hot 100 for one week last November.
Autobahn - Kraftwerk
https://www.youtube.com/watch?v=rPgaz3Re0ik
3:38 minutes as a single and the full version 22+ minutes of ummm…music? I used to like listening to both versions and unconsciously turn my head as the “cars” zoomed past.
Non-German speakers misheard “Wir fahren fahren fahren auf der Autobahn” as Fun, fun, fun on the Autobahn thinking it was a tribute The Beach Boys, but it really means, We’re driving, driving, driving on the autobahn and along with the rest of the lyrics supposedly about the monotony of the long drive.
Funny thing I was thinking about that song a couple months back and looked it up on YouTube. After finding the Singing Nun version I managed to find this gem, directed by no less than Fellini. I’ve watched this many times and I have to admit that it’s grown on me.
The mention of Saigon brings to mind Paul Hardcastle’s “19,” which was a pretty big hit in 1985. Mostly sampled audio clips of news reports about the Vietnam War, with a techno music background and a little bit of singing.
Speaking of Vietnam War-era popular songs, there’s “Battle Hymn of Lt. Calley”, which made it into the top 40 briefly and may be the only song of its era to glorify a war criminal. I own the 45, a true collector’s item.
In the category of most unusual chart success, there’s Wanda Jackson’s “Fujiyama Mama”, which inexplicably made it to #1 in Japan for several months in 1958, despite Wanda belting out lyrics like this:
I’ve been to Nagasaki, Hiroshima too!
The same I did to them, baby, I can do to you!
And since we’re picking on spoken word “songs”, mention should be made of Gordon Sinclair’s embarrassingly fawning “The Americans (A Canadian’s Opinion)”, which got a lot of airplay in 1973. At least its profits went to the Red Cross.
How did Country Joe and the Fish do on the charts?
Yeah, that’s a good choice.
The Deck of Cards, a religious glurge song that somehow not only gets high on the charts but keeps getting updated for the latest war.
When I first heard it as a kid, I took it seriously, but now I know the soldier was really playing cards in church but was just bullshitting to avoid punishment.
I Feel Like I’m fixing To Die Rag went #1. In Sweden. In Australia, #60. In the US, not so good.
In the US their best was #98 for a single, 23(!) for an album (Together, which I’ve never heard or heard of before I checked)
Speaking of spoken word (albeit in verse) hits, there’s Johnny Cash’s only top ten hit on the Billboard Hot 100 coming in at #2 in 1969.
The mere mention of Sweden brings back memories of a Scandinavian Electro Country classic:
And man, that was worldwide massive - Cotton Eye Joe (Rednex song) - Wikipedia
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Tom Lehrer is great indeed, and your suggestions spot on. But try Wernher von Braun or The Elements too. The Elements was even sung by the chemist in Better Call Saul while in his lab when Gustavo Frings came to visit him, it is nerdy popular.
He is still alive, btw. Should be 94 by now.
Does anybody remember Lydia Lunch? Suicide Ocean is just one example, she was fantastic:
Probably not very popular, I guess…
Though for the weirdest music album ever, Lou Reed’s Metal Machine Music is unbeatable.
Yes, the whole album is like this.