Check the detail screen for that villager -- nursing moms will have a timer that shows how much nursing time they have left. The babies nurse until they are 2 years old, I believe.

Did your villager happen to drink any strange liquids? There's a known issue with pregnant moms who suddenly get much younger due to an island event -- they will nurse until they reach the age they were when they drank the liquid, which can be quite a long time.

She didn't drink anything strange but she's been stuck on having 51 minutes left of nursing for awhile. Maybe she drank the strange liquid when I wasn't looking! It's a village conspiracy!

Certain berries will do that, too. You have to be careful what you let your villagers consume - some of that stuff has lasting effects!