Ok, so the brake lights work.
Now check all fuses, including those under the seat. If those are OK, can you hear the fuel pump prime when you switch on, but before trying to start? If not, then suspect the 'injection loads relay', on the LHS under the seat. If yes, then suspect the starter relay, on the RHS under the seat, or the starter push-button itself.
A multimeter or test lamp is pretty essential for quick and easy diagnosis.
With photos for reference: -
I have checked the fuses in the glove box and the single fuse in the battery compartment. All appear to be fine. I cannot locate any fuses under the seat - am I missing something here?
When I switch on and before I attempt to start I can hear what I believe is the fuel pump priming OK.
I don't have a multimeter just yet. I have physically inspected the brown relay on the RHS and nothing seems to have worked loose on there. Is this the injection loads relay? When I try to start this unit is clicking/whirring.
That relay is the starter relay. If that is acting like a buzzer, then your battery is shot, with probably at least one shorted cell. Get that multimeter! The fact the dash lights work doesn't prove a thing about the ability of the battery to maintain 12V while passing the 50A or so of the starter stall current.
This is not an uncommon situation. What happens is that there's enough juice in the battery to get the starter relay to make, but as soon as it does the large stall current of the starter motor pulls the battery voltage below the drop-out volts of the relay, so it drops out. The volts rise with the current reduced, rinse and repeat.
The failing battery can be down to too many short journeys, or being left unused for so long that the battery self-discharges and sulphates. Usually there's no way to resurrect it properly - a new battery is required.
I can't locate anything like a relay on the LHS, however there is a white unit (see photo) on the front of the under seat compartment. Is this the starter relay? It's not doing anything obvious when I try to start.
The 2015 models have obviously moved that relay somewhere else - sorry, I'm not so familiar with the layout of the newer models.
The starter-push button shows no obvious signs of damage/looseness.
The scooter is 18 months old and has just over three thousand miles on the clock. As things stand it's unlikely I can get it to the dealer before the new year so any further suggestions gratefully received. Thanks in advance for any further advice. I can almost certainly lay my hands on a multimeter if needed.
Maplins for a cheap multimeter - new battery from many sources, including Halfords.