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There's this episode of I Survived: https://www.youtube.com/watch?v=4LBajwOZmbc
(warning for description of, well, at least a similar accident to what OP described above), but unfortunately the description doesn't give the man's name or the episode number to track down the full episode. Comparing the man in the video to this article, I believe he might be John Thompson, which would be another way to track things down: https://bismarcktribune.com/news/state-and-regional/the-kid-in-the-bathtub-twenty-years-later/article_84f9bf0c-54d8-11e1-9123-0019bb2963f4.html
All I vaguely remember is this was a live-action television show...maybe an anthology of some kind? There was a an older man who I believe was a farmer. He gets his hands cut off in an accident with some kind of farming equipment and struggles back into his house and uses his face I think? to call his sister and have her call an ambulance. That's all I remember!
EDIT: now that I think about it, I'm actually pretty sure this wasn't an anthology... I think it was a true story/documentary with re-creations and an interview with the guy it actually happened to.
Edited by donatello-is-the-fellow