I like the nanny service because they sure do keep the house going while the parents are working but is there by any chance a mod that makes the nannies automatically teach toddlers how to walk, talk, and use the potty? My sims are working like crazy and only have weekends to teach the children but time ticks by fast and toddler motives decay fast and skill builds slow leaving a toddler to become cranky and basically grow up un-trained, If there is no nanny teaching toddlers mod than is there a faster skill build mod that also affects toddler skills?

No, not the game ones. There are other ways, for example use Christionlov's transport mirror to temporarily move a sim like grandma in and have her train them.
Skills can be taught very quickly via game methods if you know the tricks. Have the parents become plat from a date and get them smart milk and a thinking cap. Used together you can teach all skills in one day.

I've used a cheat that let's you control other sims, but got no option to Potty Train (maybe this mod works differently than Christionlov's). I have seen SOME nannies autonomously do it: WHY?

I personally never witnessed a nanny teaching any toddler skills, and I have a good one. Maybe the relationship score has something to do with it, maybe not.

I have only control The NPC nannies One time. By using Merola Control Mirror.
I use it to make the nannies Clean up her mess on the floor, where she pee.
I always keep that mod in my game to control who I want.

NOTE: Do Not use it On any NPCs that are Not Playable like Mrs, Crumplebottom and a few others.
NEVER control those NPCs if you don't want your game going down the rabbit hole.

Yup, I can confirm that if the relationship is not high enough not even relatives can teach the toddler skills. It happened to me with the Newson family: the brother was able to potty train etc to the toddlers while the sister wasn't; anyway, I just needed to improve the relationship until something like 15 to be able to teach the skills controlling the sister.
Maybe I've never noticed before since usually the relationship with the parents/siblings it's quite high by default. Not sure if it would work with the nannies, though.