The backing band was Reilly, Street, and Paresi and, according to Goddard, tapes supposedly exist in some form. Here's an excerpt from p. 464 of the Mozipedia (the "Viva Hate" entry):
Beyond the underlying friction between Street and Reilly, morale was further shaken when the Viva Hate group took a break to the BBC Studios in Maida Vale, London to record Morrissey's debut solo radio session for John Peel. 'A very sad experience,' recalls Street. 'The BBC engineers had a bad attitude. They were treating me like I was just a bass player so when I was trying to explain that I was the songwriter and producer they just ignored me. Morrissey was unhappy. Vini was in another one of his moods. So it was a f***ing nightmare, basically.'
The session was never finished nor broadcast, though prior to his death Peel informed this author that his producer, the late John Walters, retained a copy of the tapes. The running order remains unconfirmed though Paresi is sure that 'Suedehead' and 'Margaret on the Guillotine' were among those attempted. 'The saddest thing about the whole Peel fiasco is that it had a massive impact on Morrissey in terms of thinking whether he could play live with us as a group,' adds Paresi. 'That was a huge disappointment for me, personally, but another blow to [Morrissey's] confidence before the album was even finished.'