IIRC if/when you win you ESA claim are you on PIP as well? Any and all Cold Weather Payments (if you meet the conditions of entitlement when the payments are triggered) should be paid in full. 





It's a bit like SDP which is a premium that is payed when you qualify for it no matter how far in the past that was! I got a couple of cold weather payments a few years later a while ago with a big apology because they forgot to pay them one year so....... 





There's a class action by CPAG on ESA assessment times see here https://www.rightsnet.org.uk/forums/viewthread/16063/





The Warm Home Discount is another matter? That's a £140 notes and see my post in reply to Alteredchaos in this thread as it's a little bit different in the argument!