I used to busk in and around Nottingham in about 1983/4, playing music, but not using the harmonica. A couple of times, i did fire eating too.
Do you mean “busking” or “streat entertainment”? A busker stands there and plays an instrument or sings, or quietly gets on with a bit of juggling and waits for the money to go in the hat from people passing by. A street entertainer occupies an area of ground and puts on a show, encouraging people to stop, watch and pay.
The difference is important, because England’s laws are enforced only sporadically, and occasionally with a surprising degree of common sense. So if you aren’t blocking the main thoroughfare, intimidating anyone, creating a public nuisance, or actively begging for money, the worst you will usually get is being asked to move on. On the other hand, if you mark out an arena with a 50 footlength of rope, address members of the public directly and loudly, blow 6 foot spouts of flame, and run round the front of the crowd with a hat demanding payment (be it ever so jocularly!) you might end up upsetting a Bobby.
Then you have to be sensitive to the place. In Nottingham, it’s OK to busk in the subways (pedestrian underpasses), and on certain shopping streets. You wouldn’t get away with it in the Market Square, or outside a municipal building. If you are somewhere touristy (and I guess Oxford falls into that category), and somewhere where there may be lots of other potential buskers (e.g. a major university city…) then the rules might be enforced a bit more rigorously.
If you ask for permission and it’s refused, then you are in trouble if you get caught doing it. if you don’t ask, and you get caught, you may well get away with a reprimand. (Ignorance of the law is no excuse, but genuine well meaning ignorance coupled with a polite show of imbecility can help.)
And if there are regular busking “pitches” in your area, don’t muscle in. Some buskers do it for a living, not just for a hobby.