we live in UK so pretty used to wet weather and my husband also uses a mobility scooter
you can buy clear waterproof covers for the front console which will protect starter etc from rain
battery being under the seat is less likely to get a soaking but best to get waterproof covering for the rider which will also cover battery
my husband has a poncho type which covers his knees
waterproof trousers and overshoes which fit over his regular footwear
he uses a rucksack type arrangement which fits over back of seat to store all this stuff until required!