First - Snuggling vs. Cuddling

Snuggling and cuddling can mean the same thing which is something like "hugging" for comfort or intimacy. But "cuddle" - while it can be mutual - often implies one person cuddling another. That is, there's can be a sense of a "cuddler" giving comfort to a "cuddlee". The following is mostly described as "a father cuddling his baby":

And here, our feline friends "Odin and Ruby are snuggling". Also, Ruby (the cat on the right) is snuggling his nose into Odin's white fur. And just like Ruby, a single person can feel good just "snuggling under the covers". But one could also say, "these cuddling cats are cute".

Now the Answer

Let's assume your scene is a mother talking to her child. The mother is sitting next to her young daughter who is near a doll. The mother is offering some consoling words to the daughter, who is afraid of the dark.

A) "If you're scared, snuggle with that doll, baby." is the most natural of the three you presented. This would be especially true if the doll was rather large and already next to the child. It implies a mutual snuggling, just like Odin and Ruby, above. That would be exactly the correct thing to say to a young child, since the idea of the doll being animate and being able to provide comfort back is what provides comfort to the fearful child.

B) "If you're scared, snuggle that doll, baby." could sound a bit too curt - like an order. However, there's a type of mother-character one might see in movies (or real life I suppose).. the no-fluff, down to earth, common-sense, yet-loving mother that might deliver the message like this.

C) "snuggle up to" focuses more on the movement, and may suggests a more playful action. (Also, between adults, snuggling suggests more intimacy than cuddling.) This would be the least natural, in this case, in my opinion.

The difference between using "snuggle vs. cuddle" with a child would mostly depend on what the child already thinks of as being comforting. It's not the specific word, but rather, which word the child would feel the most comforted by. One could suggest that a child cuddle her doll (like the father above) or cuddle with her doll (like the cats) for comfort, if that's what she is used to doing and calling it.

FYI: This is also reminiscent of the Harlow Monkey Studies.

Google Ngram: snuggle *,cuddle *,snuggle that,snuggle this,snuggle me,cuddle that,cuddle this

Both sentences are possible. You are correct that using the present perfect I haven't started yet is natural and idiomatic. This is because you have specifically asked about the present time: where you're (now) up to. The present perfect allows one to connect the past action of not starting to the present moment, the now. And all that is quite natural.

However, it is not wrong to use the simple past. True, it talks only about the past action of not starting, so semantically it does not have that same "in built" connection to the present (the now or moment of speaking), but neither is it wrong to use it.

Hope this is helpful.