Buy a wall battery charger. Quite often they are sold pretty cheap (ca. 40CHF) in Aldi or Lidl. A very small investment compared to the price of batteries. It will keep your battery alive during the winter stop.
Follow epicure indications. If the battery and started are ok, your scooter should try to start. That means that the starter turns (as it happens with old cars in winter mornings). If it doesn', your new battery not charged, your starter is dead or there is an electrical gremlim (disconnected wire or whatever). Or the engine swithch is off, as epicure said.
If it turns but it doesn't start, check if the petrol petcock is turned so that no petrol gets to the engine (hey, petrol also evaporates if the cap is not right, check it out!). After that, just change the sparkplug (not all the same, check the model). And insist long, but beware for burning the started. You may check the air filter, if it is too dirt, but it may be harder.
And if you are a member of Touring Club, just call them!
Maybe I come late, but I just joined. Good luck,
Floris