I'm British, and I refer to ketchup as ketchup. This may be regional; I'm southern-English. I'd probably by default assume that if you said 'tomato sauce' you meant a generic tomato-based sauce, not the specific condiment Heinz &c make.

Also southern English, and this matches my understanding of the terminology. But yes, I believe usage does vary regionally.

I'm British and I've only ever experienced people calling it ketchup

I’m British, from the South East, and it was tomato sauce in my family when I was a kid, but now it is ketchup. There was a transition at some point fairly recently. English evolves easily.

I think you've nailed it. I'm was Essex/Kent born and raised, with East End of London heritage. It was always "tomato sauce", was very rare to hear it referred to as "ketchup".

or even worse, "Red Sauce"

Yeah, you hear that a lot in "greasy spoon" establishments, perhaps because there isn't enough tomato in the recipe to claim it as such?