After winning my appeal back in January I've got the extra payment (LCWRA) put into place almost immediately but no communication regarding back payments. The Job Centre didn't honour my ESA so I calculated something around £5,000 that was owed to me since November 2016 but wasn't certain and, although I drafted a fiery letter, I never made contact with them about it and the letter was buried and forgotten.
To my surprise the back payment arrived last month but their calculation is from February 2017 BUT I'm glad that I didn't send that letter because my calculation was wrong: they sent me £7,070

(with the majority of it gone back to my sister).