Rollup merge of #142210 - Kobzol:tidy-auto-builds, r=Mark-Simulacrum

Run `mingw-check-tidy` on auto builds

This has two advantages:
- It moves `auto` builds closer to being a superset of PR CI builds
- It allows us to reuse the Docker cache for the job in PR CI, thus speeding up the job in PR CI considerably

Discussed [here](https://rust-lang.zulipchat.com/#narrow/channel/242791-t-infra/topic/PR.20ci.20seems.20much.20to.20slow).

r? ``@Mark-Simulacrum``

Author: workingjubilee