An electrical safety certificate is required to be issued after commissioning or connection of any installation (or part installation) to the electricity supply.
A warrant of electrical fitness must be issued by an authorised person. The specific requirements can be found under sections 76 & 77 Electricity (Safety) Regulations 2010 (external link) .
In regulation 78 (external link) you will find all the details of who can issue warrants of electrical fitness.
A Certificate of Compliance certifies the compliance of electrical installation work prior to its connection and is issued by the person who did the work or supervised the work and who is authorised to certify the work.