I’ve already put that in and requested either 1 or 2 months notice period but seems they won’t accept. I have signed contract with the new company.
They can refuse to accept, makes no difference as you won't be there from the date you put on your letter!
You may end up burning bridges. You have tried working with them to find a mutually agreeable date, they are standing firm and not compromising at all - that works both ways ie this is my leaving date and you need to start looking for a replacement or how best to manage your absence.
Of course this is after all talks have broken down.