I’m a recent customer as well, my experience may possibly answer your specific questions.
The end of the month refers to the anniversary of when you switched and not the end of a calender month.
My switch was on the 2nd May, so my first bill was due on 2 nd June (which fell on a Saturday) . I received an email on 30 May to say my monthly payment will come out in 3 days time and ideally could I send them my meter readings which I duly did on on 31 May and on 2 nd June received another email to say the cost of my energy and credit since 2 nd May. Since 2 nd June fell on a Saturday my payment wasn’t taken out until 4 June (a Monday). You can see your bill by going into your account dashboard. The bill was based exactly on the readings I supplied to Bulb. Sending of emails and bills isn’t dependent on being on a Saturday or Sunday but actual payment would normally be a working day (i.e. a Monday of payment was due on a Saturday or Sunday).
You can however submit your reading now (21 st June) if your switch was on 24 th May, you don’t have to wait for an email.