What happens to TypeScript when ECMAScript brings it's core elements to the language?

Typescript for a while has focused on Stage 3 and Stage 4 proposals, so its "core elements" are almost all already in the language.

Unless you mean the type annotations itself? The good news there is that there is a Stage 1 proposal to take the Python/Ruby approach of bringing them into the language: https://github.com/tc39/proposal-type-annotations

Were that to happen, that proposal doesn't apply any type checking semantics for the browsers themselves and you would still want a type checker, such as Typescript, to actually check the types that are annotated. (Just as you want to run Mypy or Sorbet, respectively, in the cases of Python/Ruby.)