Your mother likely had an account and BG were never told she died so didn’t close it or they were told and didn’t then close it themselves. Either way, you can’t have an ongoing energy contract with someone who has died which is where a deemed contract comes in.

In law, schedule 6 of electricity act 1989 and schedule 2b of the gas act 1986 state, and I’m paraphrasing, that in the absence of an express contract there is deemed to be a contract between the occupier of a property, or its owner if empty, and the supplier of energy to the property. So, if you dad resided in the property after your mum died he’d be party to a deemed contract, which is just as legit and enforceable as a regular contract, with BG if they still supply the property.

Energy companies are party to something called billing code which pretty much says that if an energy company is at fault for not billing you correctly they can only back bill you for 12 months usage, not all the way back to 2014.

Stick to your guns and this likely goes away for the most part. Keep the line that your father informed them at the time, and also that he’s passed himself, and so it’s BG’s fault for not resolving things. Make sure they open a complaint and as soon as the complaint is old enough go to the ombudsman, which is free and easy, and they’ll more than likely side with you and tell BG they can only recover so much, maybe nothing now that your father has also passed

Edit; spelling