Hi- You poor thing-you must be shattered!
Having a cocker puppy is mind blowing anyway without sleep deprivation!!!
I'm no expert but ours is fab at night so I can tell you what WE did-May or may not work for yours but was brilliant for ours.
Before I write it, I would also say, yours has been a little bit inconsistent to start with, so whatever method you decide to go for, I would say go for it as though you have only just got pup home..start again.
We did t like this: Pup in her sleeping place but not able to get out. Me on the sofa nearby. If she whined, I would not go to her, or even speak to her properly, but I would make some sort of noise- breathe loudly/ shuffle around/ make a sound so she could recognise it was me and that I was there but that I was 'asleep' I did not get up. However, for the first few nights I would get up at 5am and take her for a wee. Then we would snooze together on the sofa until a more reasonable time. I think this helped make the nights shorter and more manageable for her to start with. I did this for a few nights and gradually increased the get up time. Once it was approaching 7am I went back to my bedroom and she was fine. The whole process took just over a week for us(would probably take a bit longer for you now as you are restarting) and for the following month we would make sure that as soon as someone was awake they shot down and let her out so she was in there for as little time as possible.
Now she will just wait patiently until someone arrives. (I should note at this point that she is not in a crate, but an area about 4'x3' so she has plenty of room to stretch and also has a water bowland a couple of safe toys In there.
In the early days I think this may have helped because it took ours quite a long time to house train and if she needed to go in the night she would be able to get out of her bed and go elsewhere in the area and be able to get back into a clean bed. Not lovely but it didn't last long and we would just clean it up in the morning with no mention of it.
We did the cold turkey thing with our Springer, and like you, he screamed the place down. In the end it wasn't a quick fix and I felt bad afterwards because the poor little thing was probably terrified and missing his mummy. I think showing them what you want them to do but reassuring them that you are still there and they are not alone is much kinder-especially for cockers who are so clingy.
You must be feeling really stressed out now, especially as you're now feeling like the kids are suffering etc but this is a really important thing to sort out. There will be other school holidays but there will not be another puppyhood for your dog. Much as it is really hard work you have to do what is right for the pup right now or you could have persisting problems. Your relatives/neighbours are just going to have to suck it up! It's your dog. It won't be forever. Have the confidence to decide on a plan and really stick to it. Even if it takes a month (which it won't), that is nothing over the lifetime of the dog. By October half term you could be all sorted and take your kids out for the day to make up for the patchy summer hols ( but they wanted a dog too yeah?) I would apologise in advance for the disruption,buy everybody some earplugs and teach your little pooch how to do bed time!
Best of luck-it will be fine in time!