I wouldn't mind cole slaw, but I hate carrots, especially raw. I like the cole slaws I have had without carrots, but a lot of people say that ain't cole slaw.
It ain't.
Call it ole slaw. Seriously cole slaw is one of these dishes that has a thousand variations. Do it however you want and just call it J's cole slaw. I once had one with pineapple in it and thought this is gonna suck big time. Turned out it was actually quite nice. For the odd occasion.
I always thought of cole slaw as a side dish rather than a condiment, though I can see how it could be used on a sandwich. I'll have to try that sometime and see. Just never thought about that. I mean why not, some people put what is effectively a basic sald on sandwhiches (lettuce, tomatoes, cucumbers, onion, etc).
And I would say cole slaw is very popular in both UK and US.