chore: upgrade to TypeScript 5.2.2 (#55105)
### What?
Upgrade TypeScript to the latest version as of this PR. **This does not affect users, as the change is only for our repository.**
### Why?
Part of some upcoming PRs to try to clean up cookie handling, now that `getSetCookie` is available. Since we use `undici`, which [implements it](https://github.com/nodejs/undici/pull/1915), we can get rid of some code to rely more on the platform.
This PR is needed to get the types for `Headers#getSetCookie` which was added in 5.2
### How?
I needed to update some dependency types to get build to pass, but other than that, only needed to bump from `5.1.6` to `5.2.2`, so hopefully all is fine.
