Backport PR #43729: DEPS: Upgrade Deps for Python 3.10
- [ ] closes #xxxx
- [ ] tests added / passed
- [ ] Ensure all linting tests pass, see [here](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit) for how to run them
- [ ] whatsnew entry

**THIS IS A MANUAL BACKPORT OF LARGE BUILD CHANGES. PLEASE REVIEW CAREFULLY BEFORE MERGING**