It's totally doable, and the shuttle bus services are really reliable. Every hotel has some sort of service. Kissimee or International Drive are easy and very well catered for for Brits (the Wal-Mart has a British section if you just can't leave the Marmite and Fruit Shoots for two weeks). It does make it a little difficult to move around in the evenings if you wanted to have a hunt around for a restaurant, but to be fair, that's probably the same if you don't drive here.
Ive been lucky and never ever had any weather problems in Florida, occasional torrential downpour, but they clear quick. I think that's me being lucky though. I've been in every month from May through to November. Have only had a car in Orlando once, the rest of the time was transport. I've been to Orlando, St Pete's, Clearwater, Naples, and I've done the stay in Disney thing (stupid expensive), and the villa thing (cheap as chips). If you are confident enough, book it all driectly, you'll save a fortune, and only pay for the hotel when you leave, not 10 weeks before. It really is easy.