You could ring Shelter for advice. There’s also a lot of stuff about tenancy rights on the Gov.uk website.
Don’t stop paying your rent because you will get a Section 8 served against yous for non payment. That can get very messy and will probably end up in court. That’s when you might get a CCJ (aka Mcol) against you which will affect yr credit rating. This could stop you getting things like phone contracts and any other kinds of credit. It can really mess yr life up.
Your LL might serve a Section 21 to get the place back. He might then get the heating sorted out whilst it’s empty? New boilers can be about £3+grand.
If your contract is still running as a “periodic” tenancy you’ll need to sign a relinquishment document, return the keys, have a final inventory taken and clean the place up according to contract before any deposit gets returned to you. If you leave the place dirty, he will use deposit to get it cleaned.
I doubt the court will make any award to you for heating bills in excess of the usual. Your court fees could be excessive to even try and wipe out any gain.
A plumber in not a heating engineer. You need to learn the difference. Some are both but I’d say that’s unusual. Sounds like he should have sent a certified heating engineer with whom you could have had a proper conversation and asked a lot of questions. Everyday’s a school day.
As others have said, you could have learned how to turn off the boiler completely at the main (think about it …. boilers have to have an off/on switch) & used other forms of heating when you knew it was running up big bills? You could have learned more about radiator valves and how heating systems work maybe? A tenant should always know how to turn off water at the main stop cock in the house and out in the pavement, how to isolate the power supply and how to work the boiler. Mr Google is always available to learn about things like that.
The bricks & mortar are the landlord’s but it’s your home whilst you’re living there. Treat it like your own and learn about how it all works.
If you bad mouth the LL during viewings, he probably won’t give you a good reference when you need one for your next landlord so you’ll struggle to find somewhere. Be careful because he might find out!
They’re two sides to every story. Good luck with moving on from this one.