It's actually from the broadway stage show, The Sound of Music (1959). As the Movie didn't come out til 1965 it would be a bit of a time warp for Coltrane to have covered it in 61 ;)
Song copyright is a tortuous, torturous & sometimes tort-urous game of legalese.
Basically, it boils down to "Once a song is written & published it can then be covered by anyone who wants to; subject only to correct accreditation & payment of appropriate royalties."
If a song is "substantially modified" from the original then special permission can be needed, but not always & it was less insisted on in the past.
Obviously, someone like Coltrane is going to fairly "substantially modify" a piece, but who would deny him permission, when sales are guaranteed? Once a piece has been significantly covered by many, then that aspect of modification tends to get ignored. By the time you're covering something 60 years after the fact, no-one cares any more. Just release the cover, pay the royalties.
As Ariana Grande's cover is a partial re-write, negotiations will have taken place as to who owns what percentage of any part of the original vs new. If you look at the credits, you will never find percentages, but you will see it has shared credit.
Songwriters: Ariana Grande / Njomza Vitia / Richard Rodgers / Taylor Monet Parks / Charles Michael Anderson / Kimberley Anne Krysiuk / Michael David Foster / Oscar II Hammerstein / Thomas Lee Brown / Victoria Monet McCants
The original composer/author team, if you weren't aware, was Richard Rodgers & Oscar II Hammerstein.