This doesn't ring true for me. Anne & I were in London together from 1958 to 1960, and were regulars at Ballads & Blues in Soho Square, which doesn't seem to be mentioned these days. As far as I recall, Long John Baldry never appeared there.
I first heard John in a coffee bar in St Martins Lane early in 1958. He sang unaccompanied one of the solo blues from the Parchman Farm recordings on "Murderer's Home". This was a straight copy, and it was near-perfect - pitch, intonation and accent.
I heard him a few times over the following year, mainly at the Partisan, and his American blues singing was always spot-on. I heard him sing a number of things, but never the popular skiffle songs.
Peggy reports the incident as about 1960. I can't believe that in 1960 Long John Baldry would put cockney vowels in a Leadbelly song - unless he was parodying the Lonnie Donegan version. It would fit with his sense of humour.