I was an ADI (approved driving instructor) for eight years, until about 2013. I loved it, but eventually I found it quite difficult to earn enough where I was living, so I changed to other freelance work.
Hours: most of the demand is after school, evenings, weekends, some daytimes possible. It can be possible to work around family, but you have to plan carefully.
It's vital to check out the competition first: some areas are saturated with instructors. The industry is very competitive, and during the recession, lots of instructors found it difficult to find enough work: I'm not sure what it's like now. People decided to become ADIs when they had been made redundant, and learners were strapped for cash, and decided to defer learning. You can check out the competition by calling driving schools and asking what their waiting times are like: if lots of them can offer you a lesson tomorrow, they're desperate for work! 
Don't tell them you're thinking of becoming an ADI when you do this, because they might not tell you the realities of whether there's enough work, but would happily charge you the fees for training and franchise.
Choose carefully who you train with. It can be difficult to get the qualification, and you're only allowed a limited number of attempts (at least, this was so when I did it). Smaller driving schools can be better for this: ask the trainer if they are ORDIT registered (Official Register of Driving Instructor Trainers), and have a white badge in their windscreen as well as the green one to show they are an ADI. Also small schools need a good reputation to survive. Some big schools charge a massive upfront fee for the training, which you have to pay at the start whether you pass or not, and are not always the nicest to work under. I know several ADI's who were tied into big contracts with them, and given hell when they left (especially if they tried to take pupils with them). They often don't tell you that the exams are quite difficult to pass, so good training is vital. Better to go with a trainer who charges by the hour.
Re the car: big driving schools might provide a car (for a hefty fee); otherwise the usual practice is to lease a dual-controlled car.
You can meet other ADIs, and make contacts, by going to driving test centre waiting rooms, and chatting to them while their pupils are out on tests.
Despite what the adverts tell you, it can be difficult to earn a decent amount: lots of competition can mean that lesson prices have to be low. Big driving schools sometimes offer deals "10 lessons for £150" which small schools can't compete with. Fuel is expensive, and car repairs can be too. I earned about 20k doing 40 hours a week; but I'm not sure how realistic this is at the moment.