There are currently no licensing requirements relating specifically to car valeting businesses.
If you play music in the workplace (or in a cafe area) then you normally need a Music Licence from PPL PRS Ltd. There is an annual fee for this which you can pay online on the PPL PRS website.
Note that you will need permission from your water and sewerage provider to discharge any effluent, such as dirty water and shampoo run-off from cleaning vehicle exteriors, into their sewer (whether it's a foul or surface water sewer). In rented premises you may also need permission from your landlord. You can find out more on the Gov.uk website. The pollution prevention guideline PPG13 included specific information on trade effluent arising from vehicle cleaning. All PPGs were withdrawn in December 2015 but you can still access them from the NetRegs website. They are a useful source of information on good practice but they are no longer kept up to date so you can't rely on them as a source of information on legal requirements.
The Car Wash Advisory Service (CWAS) (previously known as the Car Wash Association), which campaigns to raise awareness of 'unregulated' hand car washes that pollute the environment and discharge effluent illegally, has information about handling waste water on their website.