"I think" isn't going to be good enough I'm afraid. You will need explicit written permission in some form that allows you to have pets. If you don't have this your landlady is perfectly within the law to tell you you are in breach of contract and tell you to remove the animals immediately. Standard tenancy agreements have a "no pets" clause and you have to ask for it to be removed and replaced with the "pets allowed" clause or you will always be in breach of your tenancy.
I used to work in the lettings industry
Until you sort that out no rescue will rehome to you, it's too much of a risk for the cat to rehomed and it not be a forever home, they've already faced at least some upheaval and it's about keeping the stress for the cat and the strain on rescue resources to a minimum.
