There's no such thing as a 'scooter licence'. You can get a moped licence, but almost nobody bothers, because all it entitles you to do is take off your L-Plates and carry a passenger on your moped, which is not something many people actually want to do.
Otherwise, you're looking at a motorcycle licence or a car licence. Various three wheelers (most famously the Reliant Robin, and more recently the Piaggio MP3 400) can be driven/ridden with either a car or a motorcycle licence.
If you don't have a licence at all, and want a three wheeler you can drive on a provisional licence, I think the
Piaggio MP3 125 is probably your only option.
TBH, you can get a perfectly serviceable car for well under £500, so I'm not quite sure how a three wheeler is any more affordable.