Wikipedia says Adam Faith was offered it but turned it down.
The Beatles Bible doesn't mention that, but has a quote by Abbey Road producer Ron Richards where he says ...
"How Do You Do It was brought into my office by the man who wrote it, Mitch Murray, along with Barry Mason, later to write a number of hits himself. They offered me first option on the recording and played me Mitch's demonstration acetate. I liked it so much that I immediately called Dick James, the singer turned music publisher, and he signed the song up straightaway. But the acetate stayed in my desk for a long time after that. We didn't know who to give it to. Much later, when George [Martin] was pondering about The Beatles' first record, I played him Mitch's acetate. He felt that it would be ideal for them and sent a copy to Liverpool right away so that they could learn their parts."http://www.beatlesbible.com/songs/how-do-you-do-it/
That makes it sound like Ron Richards heard it and Dick James bought the publishing rights well before 4 September 1962 when the Beatles tried recording it. And Ron Richards says he was offered "first option" on the recording as well, although that's obviously not the case if both Adam Faith and the DC5 were offered it as well.
Can't find any info on the DC5 version though.