This was the response sent by Gabriel Enuezie an Immigration and Employment Solicitor:
A. No, they are not allowed to do any work whether paid or unpaid in the UK as this will be in breach of their conditions of stay.
If a person wants to come to the UK and do voluntary work then they can apply to come to the UK as a Charity Worker. A second option (although not restricted to unpaid work) is for them to apply to come to the UK as a Religious Worker.
Whichever option they choose, they must first have a certificate of sponsorship from an organization in the UK which they need to submit when they apply for the visa to come to the UK.