Using the mobility scooter, at this time of year, can be quite chilly, especially on the hands, even with gloves on. I was looking at what they wear on motorbikes, in the cold weather, and it seems to be some sort of mitts, which fit, actually on the handlebar - covering the hand, plus bar, and switches, as one, sliding onto the bar. I ordered some, they have just arrived, and they are absolutely massive. They slide on the bar, then lash onto the mirror stem, with thick insulation. Apart from the embarasing size, they look as if they will do the job.
Back in the day, on the bike, I just used to wear a pair of gauntlets, usually with a pair of silk inner gloves.