You will need to change the registered address at Companies House using Form AD01. I do not believe that there is a fee payable.
I hope this answers your question, If so, kindly rate the answer and provide feedback.
If you have any further questions, please do not hesitate to ask.
I do not believe any justification is required at all. Form AD01 needs to be completed and signed by an authorised signatory of the company.
It can take a few days to change the registered address once the forms have been received by companies house.