Q. (See above question in regards to removing partially hanging nails before washing netilas yodaim). What do you do if it is Shabbos (and you cannot remove the hanging nail)?

A. If a piece of one’s nail is mostly detached and hanging and causes pain, one may remove it on Shabbos by hand or with the teeth, but not with a nail clipper or scissors (O. H.328: 31, Shemiras Shabbos Kehilchoso 14: 54). Mishna Berura (ibid. 96), quoting Peri Megadim adds that one should remove the piece of nail carefully, ensuring not to cause any bleeding. However, if the majority of the nail remains attached, it may not be removed on Shabbos, even by hand and even when it causes pain.
Horav Shlomo Miller’s Shlit’a opinion is that Mishna Berura (ibid. 99) follows the last opinion of the Shulchan Aruch and even a mostly detached nail, hanging upwards or downwards, should not be removed even by hand and even when it causes pain.
The Rov added that in these cases, when you cannot remove the hanging nails for netilas yodaim purposes, you may rely on the opinions that are lenient and wash hands as they are. (See similar in Lechem Vesimlo 198: 22 and Piskei Teshuvos 161: 5).

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a