At the beginning, some British musicians, like Tony McPhee, did faithfully copy the American blues musicians. At the outset, Eric Clapton was quite faithful, but he tended to play loud. I heard him say, in an interview, that he decided to put the blues in a 'modern context'. I think he was talking about Crossroads. Cream still had a definite jazz influence, with Baker and Bruce from Graham Bond.
The Kinks also began as faithful copies, but thankfully played on their Englishness.
Jagger and Wyman were good at imitating and, unlike the blues-rock bands, had no intention of cranking-up the blues. I'm not sure what happened with them. They sounded like disco to me at one point.
Like Clapton and Jeff Beck (now there's a jazz man), Led Zeppelin took the slow blues and belted it out! Most heavy rock bands took their cue from Led Zeppelin.
What happened is that the British musicians tended to be brought up in the middle-class suburbs of the capital, not the cottonfileds of the deep south. While postwar Britain was far from easy, they could not fully relate to the background of the black American musicians.
Trad jazz was popular in England in the early sixties and Pete Townshend, among others, said it was an influence. Church music would have had a profound effect and so too did Elvis Presley and Buddy Holly.