Sam Mendes has recorded an interview for BBC 6music which is due to air over the weekend; a clip was played today however, in which he detailed the process by which Spectre ended up with two competing theme songs.
Essentially, it seems that Radiohead were his first choice, and he persuaded Broccoli to go along with it. He asked Radiohead to submit a song, and they agreed and gave him ‘Man of War’. Mendes thought it was great (and it is pretty Bond-y), but he didn’t realise that it was a track they’d previously written, and the Bond films require an original composition. So they had to reject Man of War; but Radiohead asked if they could submit something else. Mendes said yes, but in the meantime they asked Sam Smith to come up with an alternative theme, and everyone liked that. Then Radiohead came back with their ‘Spectre’ and Mendes had the task of choosing.
Ultimately they went for Smith’s, but Mendes told Radiohead he definitely wanted to use their song. He tried to squeeze it into the incidentals for the film, but ultimately it just wouldn’t fit (I get the feeling from what he says that he didn’t just want it to be playing in the background on a radio or something, but actually part of the score.) He said it would overpower the scenes he played it over (Yorke’s vocal being too powerful) and he had to go back to Radiohead and tell them that it hadn’t worked out.
You can listen to Mendes describing this himself at about 1.40min here: