I think "stay warm" is quite common to say in America. Like if you're shopping in a store and you're leaving, it's not uncommon for a worker to say "stay warm" or "stay dry" (if it's raining) as a guest is leaving.  
What you said (be careful not to catch a cold) is also not strange if someone is going outside, but sounds more like something your mother would say.  :P

I think " keep yourself warm" is a little awkward, though. I would use "keep/stay warm" and omit "yourself".  :)

In the UK, I don't think there is an exact equivalent.  You could say "wrap up warm" at the end of a conversation.

There are some more general ones like:

"Look after yourself", or "take it easy" (more informal and street)

We have some funny expressions in other contexts: e.g. Night, night. Don't let the bedbugs bite.

Or a mother might remind her son, "don't forget to wash behind your ears" when saying goodbye. My aunty still says this, though more in jest.

Stay warm.

Drive safely.

Take care.

Stay safe.

These are all common sayings in the US when parting.  :)