You buy them. AFAIK there are no schemes or grants available, although I'm sure other people will correct me if that's wrong.
Some energy companies will fit them or you can find a company to do it. Either way, you pay for the charger and you own it. They cost upwards of £1000 installed.
In theory you can disconnect it and take it with you when you move. Most people don't because it's a hassle to remove and a lot of the cost is the installation. You'd pay to have it removed then reinstalled when it wouldn't be much more to get a new one at your new house.
Home EV chargers use something called a Type 2 plug. This is a universal standard for AC charging so all electric vehicles can be charged with a Type 2 charger.
The charger doesn't care which electricity supplier you use. The only time it matters is if you want to use an EV specific tariff that requires the charger to communicate with the energy company. They do this to trigger charging at specific times of day. Each energy company has a list of compatible chargers, and sometimes compatible cars as they can be part of the communication mechanism.