I'm not well up on modern qualifications (other than 17th) so I presume you are now a qualified sparks on paper but lack some pratical experience .
To become self employed , if I remember correctly , just inform the National Insurance people & your local tax office , job done .
When you say " Register for Part P" that means nothing ...Part P is a section of the Building Regs appertaining to domestic electrics.
So, if you are doing domestic work it would be a good idea to become a Registered Domestic Installer by signing up with one of the providers such as , NICEIC, ELECSA, NAPPITT, and the cheapest ...STROMA . This allows you to sign off your own jobs for Building Control .
If you do not intend to return to site work you won't need an ECS card or a CSCS card or a union card or a JIB card , but to be honest if you have a JIB card it would be a good idea to keep it going .
How confident are you in working alone on domestic stuff . ?