Here is the solution:
If existing symlinks belong to python 3.7 you should unlink them:
brew unlink python
Basically all you need to do:
brew link --force [email protected]
OR force the link and overwrite all conflicting files:
brew link --force --overwrite [email protected]
OR if needed list all files that would be deleted:
brew link --overwrite --dry-run [email protected]
Thus you can switch to any python version available in the Homebrew repo.
Also check out this answer for pyenv usage