The names of Leiber & Stoller were never known in the U K really till Elvis made Hound Dog and the Coasters made Searchin' a year later.With no one exploring the roots of rock'n'roll till the late 60s that's how it remained.
Leiber & Stoller had the knack of writing in the style of a black man and most of the time they took from the general stock of 12 bar blues and its variants.Never especially noted for creating great melodies like Lennon & McCartney the songs would frequently have additional output from other writers but the main thing about Leiber/Stoller songs was the lyrics.
Songs would be retreaded and in some cases were "borrowed" from other sources.Thus Stand by me was according to its part writer Ben E King "loosely based on a Gospel song" and even the title was kept making it a different song to the one Elvis recorded.
Eventually Leiber/Stoller tired of writing R & B and RnR and by 1959 were moving closer to what became high school pop or straight ahead Tin Pan Alley stuff as recordings by Peggy Lee and Johnny Mathis testify.
They dabbled in girl group pop and even in a sense the British Invasion when Stealers Wheel charted in the 70s when America looked to Britain for new trends.
Like Buddy Holly they were interested in the idea of rock'n'roll with strings and the Drifters' There goes my baby moved R & B into Uptown Soul