@kit
Forgive me but I am confused over the dates in your post.
You say you claimed UC in October 2021 and that your ESA ended in January 2022 but that you have only been awarded LCWRA in August this year.
Did you have an award as opposed to being on the Assessment Rate of ESA before the claim was closed. If you did then that award should have been reflected in your UC eith from the date the ESA became payable or from the date you claimed UC if the ESA pre-dates the UC.
But, to answer your specific question a new LCW or LCWRA award should be payable from the beginning of the 4th period following the date that a Fit Note was presented.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems