Right OP I had a baby like this - he didn't sleep night or day. I was genuinely concerned I would die from exhaustion. I was a relatively young mum and I often think if I'd been a bit older I might well have done!
I felt like I was failing too - I wasn't. His sister slept through almost from day one - that wasn't because I was good at it by then it was just the way they were built.
I have driven round at 3am with a screaming baby, sobbed pushing a park backwards and forwards in the hall, found myself staring into space with exhaustion. It's AWFUL but it does pass. Honestly. I know one more day feels like an eternity though right now.
My advice is :
Stop worrying about nursery - cross that bridge when you come to it.
Keep trying with various dummies. It might not work but it just might.
Try winding him frequently esp by sitting him forward on your knee and sort of swivelling his bum around.
Do skin on skin a lot.
Stop counting hours of sleep versus no sleep. Having knowledge of it doesn't make anything any better or easier to cope with does it?
Do try the pram (sorry!!!) just standing still rocking back and forth. No eye contact, try music or white noise if you want.
If he'll cuddle / feed to sleep then do that and safely co sleep if you can.
Get out of the house as much as you can - meet up with friends / go to the park / shops whatever. Leave the house.
Meditation app even for 10 minutes a day to focus on your mental health
Remember his sleep or lack thereof isn't a reflection of your bond or your parenting skills. Babies are different just like adults. I have terrible insomnia, dh sleeps like a log, 🤷🏻♀️
You'll get through this - please try and just take a few minutes and think of starting afresh and trying the things I've said above even if you've tried them before. Commit to giving them a fair go again. Babies change so quickly you just never know.