The area in and around Kings Wharf would be very easy to navigate in a scooter. It's not a terribly long walk to the ferry pier. I've seen folks with regular scooters/mopeds bring them right on the ferry. If you can walk onto the boat with your cane, I'm confident that they will assist you. The residents are very accommidating to the tourists.
Hamilton is a well developed city and also not too bad to get around. St. George's is a little more difficult, but not un-doable. Even if you just ride the ferry there and stay in the small area near the pier, it's a lovely boat ride with comanding vista of the island.
The combine ferry/bus passes are very reasonable. There is also a tram that runs around King's Wharf all day, so if you're feeling well enough, take the tram to the bus and you can ride all over the island and have a tour without leaving your seat.