My nanny was just standing there twitching and doing nothing. So I made her selectable and checked her queue. Feed toddler was popping in and out of her queue and then hold baby. Even if I made her do it, it popped out of the queue. My sim, on the other hand, had no problem doing any of these things. I tried another household and that nanny was broken as well. I got rid of all my CC and tried the house again and she's still standing there twitching. Forcing an error on her and resetting does nothing. I turned testing cheats on, but she wasn't throwing an error, so I couldn't get a log.

Finally, she tried to take out the trash. And dropped it and put it back in the bin. And took it out and dropped it and put it back in … rinse repeat. Then she just stood there twitching again. My sim had no problem taking out the trash.

Help.

Since this doesn't happen in a default game it must be cc related but since it happens with all cc removed it can't be cc related.

As I am privileged to know that you have been experimenting with nannies feeding babies and cleaning up baby bottles lately I know it probably is cc related. Not everything is removed/changed straight away on a lot when it is removed/changed in your download folder, things like slot locations are saved with an item on a lot so often such changes require replacing the item for the changes to take.
Op-codes that are playing when a lot is saved are also saved with the lot so changes aren't always reflected in the game straight away. Resetting the nanny should have fixed it but obviously didn't, I suggest (with cc removed) you check for and remove any baby bottles on the lot and replace the family's fridge(s) to see if things revert back to default.

Thanks. I tried replacing the fridge and that did nothing, then another nanny showed up. I eventually realised that the first nanny was not on the job at all (I could only ask her to leave, not dismiss her). I did ask her to leave and the new nanny was fine. I suspect it was because the lot had reset and she got bugged (I had pulled some hacks).

NPCs can exhibit some really weird behavior if they're on a saved lot that's been reset. If mine are acting odd, I usually clear them off with one of the teleport mods (SimBlender, Inge's cat/shrub, etc.) Sometimes also using Pescado's batbox to nuke sim schedulers can help if they're being particularly troublesome. Nuking the sim schedulers cancels all scheduled NPCs (nannies, gardeners, maids, etc.), so you'll have to rehire them.