To post an update (which has also been shared in the helpdesk), this appears to be a bug with the Next.js Runtime. For anyone willing to use Background Functions, you can use it by using standard Netlify Background Functions instead of Next.js API routes. The fix for the bug is still being investigated.
Any (recent) Next.js version should work fine. However, if you’re talking specifically about Background Functions not working, that’s not even a Next.js feature. That’s a Netlify-specific add-on, and it’s currently having some issues - as you’ve noticed. It should be fixed soon.
If you’re having some other issues with Next.js on Netlify, feel free to let us know.
Could you confirm if you were accessing the URL as /api/<route-name> and not /.netlify/functions/? The former should work as a background/scheduled function, the latter would not. We’re working on improving this so it can be more obvious in the future.