Success at last: Historic ESA > UC > LCW
Thanks to @Hart86 who advised me to leave a message, regarding an MPG1, under the Payments category of my UC journal.
After a week of inaction, I prompted my work Work Coach about it who did a "handover", also telling me to call the helpline (his favourite turn of phrase).
Called the helpline the following day and spoke to someone called Steve [long story short] who could see my ESA had been closed down on March 28 and reinstated on March 29. He said I was owed a backdated payment for LCW of £504.44, paying it straight into my account there and then, saying it would continue monthly at £126.11. When asked why it had taken so long after talking to various other people, he apologised and said he didn't know why it had been overlooked as it was a simple action to perform.
Resident joker - not to be taken too seriously ¯\_(ツ)_/¯