Regardless of which British scooter this turns out to be (and L.A.B is probably right) the fact it has a Villiers engine makes it very easy to sort out.
The basic engine is standard Villiers but usually enclosed in a cowling with fan cooling seeing it sits under the seat out of sight. Some variants have an electric start, commonly a Siba Dynastart and this about doubles the complication of the very basic Villiers electrical system. (Still idiot simple by todays standards.)
Good news is there is lots of information available, Villiers spares are not a problem and the electrics are very amenable to upgrades using modern solid state components which makes everything very reliable.
Tell us the engine number and we can go further.
Cheers,