In certain cases (like what you see on that post) the toenails are still exposed so trimming would have been possible. In cases where the toes weren't so available (which would not have been typical, given the way that the foot was bound) they could still be pulled away from the rest of the foot.

Mothers would have helped take care of foot hygiene for yet-unwed women. Husbands would have helped after marriage.

It wasn't too uncommon for toenails to fall off over time. Infection wasn't too common (a fifth of the few thousand that were interviewed by one researcher), and major blunders were even more rare. However it wasn't unheard of for toes to be removed, though this was not the norm.

For more, see the following. Be a tiny bit wary of the second source I'm listing.

• Gates, Hill. Footbinding and Women's Labor in Sichuan
• Hong, Fan. Footbinding, Feminism and Freedom: The Liberation of Women's Bodies in Modern China

More on the period when it was falling out of fashion:

• Bossen, Laurel (et al) Feet and Fabrication: Footbinding and Early Twentieth-Century Rural Women's Labor in Shaanxi
• Drucker, Alison R. The Influence of Western Women on the Anti-Footbinding Movement 1840-1911

Some of those are older sources just because there really aren't many people left with bound feet, and much of the scholarship on the subject just wasn't done recently.

If you have other questions let me know. I know you're really just asking about toenails here.