Pete Best was out before the recordings. Even though it is said that George Martin asked for it, because he thought his playing was bad, it must at least have been a joined decision with the rest of the band and Brian Epstein. Something else was going on, still unclear to this day. As a proof to this, George Martin had brought a session drummer for the first recording session of the group, Andy White, just in case. He never thought of Ringo as a good drummer, and he has stated so even after the success. During the first 17 takes of Love me do, Ringo was nervous both with the recording (it was his first one) and with the fact that there was an other drummer waiting behind the scenes. He obviously thought of the Pete Best situation and was troubled. When they went out for a break and returned to record the B-side, P.S. I love you, he found Andy White behind the drums, and he was given a set of Maracas!!! Then when they got it right, Martin asked them to play again Love me do, and Ringo was given a tambourine (or something). But it turns out that the single version has the Ringo drums on Love me do, while the Please Please me album version of the song has Andy White's drumming!
By the way, the song was written sometime between May and July 1962, after they got the phone call from Martin that they will record them.