You are not alone OP, sleep deprivation will f you up.
Immediately, call someone - friend/family/colleague/neighbour/anyone
Get some help - GP, health visitor, midwife etc. Call 111 if you are struggling to get anywhere.
Secondly - medium term solution, put something in place for the short to medium term. People you know that can help, if even for a few hours. Pay someone if needed, someone said a night nanny (not sure what this is but possibly expensive, but I would say worth it), even a local babysitter, much cheaper option and they can help for a few hours a day/night
Thirdly - definitely look into sleep training. Just pay a sleep consultant, don't try and figure it out by yourself. It works. You need to address the root cause so this can stop. Your baby is probably equally exhausted.
Given how you're feeling, please be honest and reach out to someone to help you with the above steps. It's surprising you're even functioning after this long. Get help. Any help. Be honest with what is going on and how you are feeling. Dont do it alone. You will be ok, you just need some sleep.