Although they could, it is not advised as you won't be issued a Building Regulations (Part P) compliance certificate.
Answered by Dave, Electrical Safety Expert
Although they could rewire your house, an unregistered electrician is unable to self-certify their work and therefore won't be able to supply you with a Building Regulations (Part P) compliance certificate, which is required by law for all notifiable work.
Notifiable work is a term used to describe particular types of work that are subject to compliance with Building Regulations (Part P) notification or approval. A rewire falls into this category. We recommend that you use a registered electrician for any work in the home, please visit our find an electrician section.