BUG: Fix inserting of wrong-dtyped NaT, closes #27297
- [x] closes #27297
- [x] tests added / passed
- [x] passes `black pandas`
- [x] passes `git diff upstream/master -u -- "*.py" | flake8 --diff`
- [ ] whatsnew entry

Needs tests for PeriodDType case.  The tests are probably not in the ideal place, how to move/parametrize them depends on what axis we want to sort them along (i.e. all Series indexing tests together or all timedelta-insertion tests together)