With their little cherub faces, sometimes we forget how manipulative young children are!
Your child wanted to get up at half four, so when you said no the first time they started to manipulate you... crying, screaming, banging on doors...anything to get Mummy's attention. Obviously you have a child with alot of willpower and stamina (still fighting after an hour and a half), however instead of sticking to your guns, tiredness made you compromise...completely understandable but a big mistake. Now your child knows that if they scream and bang enough mummy will give in and they get their way!
Unfortunately the fact that you gave in even once, means that it'll take more time and effort to fix! But all is not lost... you just need to dig deep and not let them win, no matter how tired or drained you feel.
First time they get up, say ' no, it's time for bed now' and put them back in bed.
Second time, just say 'bedtime' and take them back.
Third time (and every time after this), say nothing, just put them back in bed.
This will be extremely hard for the first few nights, they will probably fight you until they pass out, but don't give up. Soon, they should get the mesage that you are serious and that they're wasting their time and energy!
Putting in the hard work now and letting the little ones know that mummy's word is final will make life easier in the long run!
Hope this helps!
*We do not inherit the earth from out fathers, we borrow it from our children*