right, first stop is the homeless people and explain you are "soon to be homeless". i was in exactly the same situation...not eligible because of my partner. kind of a catch 22.
has there been any domestic violence? if so, that gets them moving quickly. they take it very seriously. i had never pressed charges against mine, but they believed me anyway.
another thing that helped me is that i had already begun to stay other places, as my home was unmanageable at that point. i had stayed with some friends a few times...they did ask me if i had stayed other places at the homeless office...i went and stayed with a girlfriend who was already overcrowded with 4 kids in a two bedroom.
the other thought that comes to mind is a budgeting loan. there is eligibility if you have recently separated from your partner. they'll give you £330 to get on your feet.
naive