The police have very little idea as to the various intricacies of busking in the UK.
The reality is that there is no law in the UK that makes busking illegal, there are bylaws which prevent busking in certain places, but these are few and far between and only affect that particular piece of land. Interestingly, this type of bylaw also includes those preventing market research surveying, and leafleting in the same areas, but nobody believes that these activities are illegal.
EG:Gray versus the chief constable of Greater Manchester in 1983,
Gray, "A street musician who played the guitar in a passageway and was given coins by passers-by, was held to be not guilty of the offence of placing himself in a public place to beg or gather alms created by Vagrancy Act 1824 s 3. His conduct did not fall within the section because (i) he gave value for money, and (ii) passers-by were not forced to deal with his activities".
This was also quoted in the law report in 1983.
(I'm not a busker, this was a quick Google search.)
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