Welcome Black Duck!
First of all, the standard disclaimer, i am not a vet or any real expert of any kind. I have however kept cats, rabbits and guinea pigs for a while. So take my opinion for what it’s worth.
Cats - yeah they seem to like cosy places but remember they do have fur and are generally quite content being outdoors even in pretty cool weather. Like most furred animals they have the ability to vary their fur due to seasonal changes and will likely adapt to your new standards.
Rabbits & Cavies - We used to keep ours in a shed outside all year round. Scotland isn’t quite as misebale as some folks reckon but i guess we probably got down to about a couple of degrees below zero (celcius) outdoors. I have no idea about the internal shed temperature but it wasn’t heated other than by the occupants themselves so i’d be surprised if it didn’t get down to about a few degrees above freezing on some nights. No problems although we always made sure they had extra generous bedding material during winters and we had several who could huddle together for warmth if needed.
Bottom line - i’d be surprised if your animal health suffers before yours does. How low are you talking exactly?