Try poking the WD40 spray tube to the back of the starter button. ....then with key off push it several times. Also squirt some in key slot....work it back and forth without starting.
Wet has gotten to a connection, bulb seat, switch, etc.
The 2 Battery post connections nice and clean and snug?
Do tail and brake lights work?
Glass tube or square plastic fuse?
Fuses cheap at any autoparts shop. Take with, if on doubt.
Age of this scooter?
(thought knickers were female undies in UK? - according to female friend from Hull - no idea why she shared this)
Stig