I’ve noticed a couple other users on Big Sur are having this issue, and it seems like maybe Local isn’t cleaning up the Nginx processes:
Can you try killing those processes and then starting Local?
Using the output from above, you should be able to use those PIDs like so:
kill 13025 74091
If the computer has been restarted, those processes might be different. Can you give that a try and let me know how it goes?