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If I change a function signature, then my code is fine - but all the other files which import and use my function will break

That's a job for TypeScript, not eslint.

Linter rules can rely on the type system

What eslint rule would apply to the caller of a function after that function's signature changes that wouldn't also be picked up by TypeScript?

In particular, the call site itself hasn't changed, as this thread assumes the linter is only run on changed files

Anamexis has a couple of examples in this response: https://news.ycombinator.com/item?id=38655101