I meant to reply to this when I saw it a few hours ago. I think it's an excellent question. This might be a bit all over the place, because I'm typing out my thoughts on the fly:
First, a caveat: You have to apply rules with some intelligence. For instance, if you're learning a Chopin waltz, it will sound absolutely terrible with no pedal, so you should pedal each measure or something initially so that it at least resembles the piece, even when just learning the notes and rhythms.
"Learning everything at once" vs "learning in layers" -- They seem contradictory, but in my experience aren't. The idea is to learn as much as you can comfortably take in at once while minimizing "mislearning". This adjusts dynamically based on your level. Perhaps a dumb analogy, but think of cooking. While cooking, you do everything at once when you know how to cook and multitask -- but while starting out, you pay attention to individual aspects. Your goal is to eventually reach the point where everything can be done at once, but you start out simplifying and working on smaller parts that you can manage. How small each part gets depends on how much you can manage at a given moment. If it's too simple, you do more of it at once, and vice versa.
For every piece, there are more important and less important aspects. This will change from piece to piece, but by and large this will almost always hold:
notes, rhythms, articulation, basic phrasing and correct gestures
come before
dynamics, pedaling, and more detailed musical ideas
I have found that your practice should be "musical". There are different levels to musicality, yes, but even when you are first learning the notes and the rhythm, you must ask yourself what the music is saying and where the phrases are going. Learning the notes like a typewriter is a real mistake imo. You would then need to unlearn many habits to bring musicality back into the piece.
Coming back:
notes, rhythms, articulation, basic phrasing and correct gestures
If your articulation is totally wrong (staccato as opposed to legato, etc.) then you would obviously need to relearn it later. So in order to avoid that, you want to get in the ballpark of the right articulation quite early on. Basic phrasing is also important: for instance, should you count ONE-two-THREE-four or one-TWO-three-FOUR. Stuff like that forms the basic skeleton of the piece.
Correct gestures are important to learn early on. Of course, there can be endless refinement of your gestures, but you want it to be quite close to what you want in the end, preferably the same motions that you would use at tempo slowed down if necessary. Work hands separately if you have to in this case; there's no real harm to doing so. But if you practice hands separate, you will ALSO need to practice hands together because hands together synchronization is also something that needs to be learned.
Hands separate vs hands together: If you find it possible to keep track of everything hands together, then do that. Otherwise, hands separate is fine in the initial stages, but you do rather quickly want to move on to hands together. You might move on to hands together practice after the hands separate parts are learned to a basic level and you can play through them. When you coordinate motions hands together, you are controlling more coordinated movements at once, so the individual coordinated movements need to be more automated so that you don't develop counterproductive habits. (Kitchen analogy: You'll set something on fire if you have all 4 burners on at once on your first day!) Even after you can play hands together, you might return to playing hands separate if a specific thing is causing you issues.
dynamics, pedaling, and more detailed musical ideas
These are typically added later. However, if you are working on a piece where the more basic aspects are immediately evident, you could directly start to focus on these aspects as well. For instance, if I were to start working on an easy Mozart sonata, I would be thinking about incorporating dynamics and pedaling in from the very start, and only leave the more subjective interpretive ideas for later.
In short, take on as much as you can comfortably manage in a given moment. If you feel utterly overwhelmed trying to coordinate both hands and the pedal, simplify: Omit the pedal, play only the bass note in the left hand and the melody in the right hand. Or simplify further: just focus on a scalar passage in the right hand that is causing you issues. You want to try to be in the "optimal zone" where you are managing as much as you can, but not so much that you lose control.
The more you can do right from the start correctly, the quicker you will grasp the piece overall, but this is predicated on the idea that you are simultaneously controlling the individual aspects correctly. So, masters can sight read a piece (do everything at once) and already have it at 80% and for them, that is the most efficient method. But it wouldn't work for a beginner.