Yes, but the bike needs to be registered, insured, MOT'd, taxed etc.
It can be done, at least one person has done it by putting their home built, high power assist, bike through the Motorcycle Single Vehicle Approval testing and then managed to get it road legal.
The category it fits into, provided the power level is kept down and the speed is limited, is Low Power Moped, I believe. If it can exceed the maximum limit for that category I believe that it has to go in the Motorcycle category.
If you Google "Motorcycle Single Vehicle Approval" (or perhaps "MSVA") you'll find chapter and verse on this.
Jeremy