You have to get your timing right, as soon as you get your ESA payment apply for UC.
There is usually a 5 week wait from completing UC and your first payment.
your assessment period is between two periods, for instance 11 July and 10 August, you will then receive your UC allowance 7 days later (17th of August) hence 5 weeks.
Gary
Nothing on this board constitutes legal advice - always consult a professional about specific problems