Yes, Motability will supply and fit a home charger for free.
Charging costs will just be part of your bill. There should be an app for your charger that will allow you to see a breakdown of your charging costs so you can deduct this from the bill you share.
If you haven’t already, it’s probably a good idea to get a smart meter fitted so you can consider signing up to a cheaper charging tariff such as Octopus Go.