BS7671 is that also known as a Part P?
No, BS7671 is a british standard entitled "Requirements for electrical installations" and is the IEE / IET wiring regulations adopted as a british standard. Part P is a section of the building regulations covering electrical safety, in domestic premises only (and other places where the supply is shared with a domestic premises, so the pub where the landlord lives upstairs, etc). Part P itself covers all domestic work and requires that the work be done in a safe manner, working to BS7671 is the de-facto way to meet this, but it does not preclude others (for example I know of someone who wanted to install something not permissable under BS7671, so he did it to the German national standard instead, and certainly at the time it would have been difficult for anyone to say that a standard from another EU nation was a problem, although I do understand that he was asked to translate the certification into english!)
Separate from this, is that Part P makes certain works notifiable, anything involving a new cirucit, replacement consumer unit, or extra points in a bathroom is notifiable, it did at one stage make extra points in a Kitchen notificable, and for some reason the amendment to take this out only applied to England, so that is still notfiable in Wales, also note that Part P never affected Scotland at all, its different in the three countries of the UK mainland!
As to what certification you should have, you should have a certficiate based on the ones in BS7671 thats relevant to the works done, either a minor electrical works certificate (MWC) or electrical installation certificate (EIC), the EIC will also include a schedule of inspections and and a shedule of cirucits. The MWC would be just a single page docment.
If the work is notficiable, you should also have a building regs completetion certificate, as others have said, depending on how the work was notified will affect who its come from, directly notified work would result in one from the lcoal authority, while being notified by through a certification scheme by the electrician would result in one from that organisation (NICIEIC, NAPIT, ELECSA, STROMA, ETC
You'll probably find that while the electrical work for your project is probably not in its own right notifiable, that because the project as a whole is notfiable that the electrical work is expected to be covered under that building notice
Sorry if thats too indepth, not t sure I can write a TL
R summary!