Reptiles are not vulnerable in the way in which you seem to be thinking when they shed their skin, so I do not see why dragons would be.
Basically, the new scales grow under the old ones, so it's not as though, when they shed that reptiles are softer during that time.
As to how long it would take, that would depend on the process. Snakes can take two weeks...
Lizards: Iguana, Geckos, Chameleon…
Lizards shed their skin by parts, being one of them the change of
colour: the skin get paler on the bits where it starts to fall off.
Skin renovation usually starts on the head or along the spine and
regularly ends on the tail. Although each individual has its own
shedding rhythm, it usually takes a week or two for lizards to have
their skin wholly shed. SOURCE
Snakes have a rather particular shedding process. Their skin comes off
all in one piece and turned inside out, shaping into a sort of pale
skin tube. Before the process begins, most snakes start presenting
hazy, whitish or bluish eyes. Colubrids tend to get a lighter colour,
while others, such as the boidaes, turn into a lot darker. The skin
shedding process will only set off once the eyes return to their
regular colours, which is usually the next day. SOURCE
Alligators and Crocodile
Alligators and Crocodile skin is scaly and often will come off in
individual scales instead. So, while alligators shed their skin like
other animals do, their “molting” is much different than other
reptiles. Just like most animals including Mammals, a healthy
alligator will continue to shed its scales regularly, rubbing up
against trees and rocks to rub off the dead skin.
As Alligator shed their scales, newer, larger, and denser scales are
formed as the Alligator grows in age and size. This makes their skin
incredibly dense and strong, resistant to the abuse that would tear or
scar the skin of most mammals, including the false theory that their
skin is “bulletproof”. While not bulletproof, alligator leather is
extremely durable and this makes it a popular leather to use in
products and upholstery where a very dense yet attractive leather is
Vulnerabilities will depend on how you work it--they might not be able to move as well, like lizards sometimes, or their eyes might be compromised for a short time. Dragons are likely more in line with gators than they are lizards, though you could work their first large molting or shedding like a lizard when they are transitioning to adulthood, and then do the rest like gators. It's fantasy, so you can do whatever serves your narrative best.
The idea of your dragons not having scales at all if they don't eat minerals is pretty darn contrary to biology. While it's fun fantasy flavor to add to the scales depending on what they eat, it would be pretty strange for them to be scaleless--they either have them or not.
Now, there are scaleless snakes--but they don't develop scales later. And scales, biologically, are there to help with a number of things including temperature regulation. While there are scaleless snakes, this is a mutation, Even the snakes considered scaleless often have ventral scales, along the belly, helping with their movement and locomotion.
Scales are made of keratin, and can be super tough. I can see the replacement scales having more minerals in them as a biologic process as they consume minerals, but even on a fantasy biologic level I don't buy them being completely without scales when they are young--the regular scales are the blueprint the mineral-filled ones can show fitness.
The closest biologic template I would say works is flamingos. They have feathers no matter what, and are born with grey feathers but their diet changes the color to pink. They don't not have feathers until they eat what changes them.
So your dragons will always have scales, but not every scale will have the minerals or some will have more minerals than others. The minerals are added to the template structure as in fish like koi or birds like the flamingo.
If you want a mineral example, look no further than humans. Colloidal Silver is used as a treatment, and it turns people a greyish blue. We have skin with or without the silver, but it tends to gather in the organs and stay, one of those happens to be skin.