According to Wikipedia, it looks like the biggest hit version of Til There Was You was recorded by Anita Bryant in 1959 and reached #30. Peggy Lee recorded a version that reached #30 in the UK in 1961.
Wikipedia has some interesting background about the version by the Beatles. Paul McCartney heard the Peggy Lee version when he was growing up (McCartney did not know it was from The Music Man until much later), and it was one of the songs they sang for their (unsuccessful) Decca record company audition. They also sang it on Ed Sullivan.
I think it's great that Ariana Grande is showing that Rodgers and Hammerstein can still be relevant for today's younger audience.