When do you have to register as an employer?
Before the first payday, not after it. Registration is free and takes a few minutes online, but the PAYE reference comes back by post and can take a couple of weeks, so leaving it until the week you want to pay someone is how a first payroll runs late.
You need to register once you employ anyone paid at or above the National Insurance lower earnings limit, anyone who has another job, or anyone receiving a pension. In practice, if you are paying a real employee a real wage, assume you are registering.
Paying a director — including yourself, in a limited company — counts. A one-person company drawing a salary is an employer with a payroll obligation, even though it feels like nothing of the sort.