Busking is NOT against the law: however there is plenty of law that the authorities can use against you if they decide that you are a problem

Causing an obstruction, or behaving in an abusive or threatening manner towards members of the public can of course get you into trouble with the police.

In addition, according to Bath and North East Somerset council:-

Under the Environmental Protection Act 1990 noise in the street from musical instruments may be a statutory nuisance and the Local Authority is able to take legal action to abate the nuisance on prohibit or restrict its recurrence; this may include the seizure of musical instruments and amplification equipment.

However, if you stick to the council's Code of Practice you probably won't run into any trouble.

The Code of Practice

1. Noise (for example music and voice) should not be so loud that it can plainly be heard at a distance of 50 metres. Busking must not be intrusive or a nuisance to nearby premises.
   
   
2. Amplification equipment is not permitted.
   
   
3. An entertainer should not perform for longer than 1 hour in any location and should not perform again in that location (or within 50 metres of it) for 2 hours. Note: particularly noisy entertainers (e.g. bagpipers or drummers) should not perform for longer than 30 minutes.
   
   
4. Entertainers shall not obstruct the highway. Entertainers are also responsible for ensuring that the highway is not obstructed by people gathering to listen or to watch the performance. The highway includes any pedestrianised areas.
   
   
5. An entertainer should not locate within 50 metres of another entertainer who is also performing.
   
   
6. Entertainment is only permitted between the hours of 10:00 hrs and 20:00hrs (10:00 hrs and 22:00hrs in July and August).