It definitely varies. But first about the molt. Some hens are fast molters, some slow. You can have a lot of variation in exactly how fast that is, it is controlled by genetics more than anything else. It’s about how fast the feathers fall out, not how fast they grow back. That can vary by individual hen, even if they are the same breed. That explains what you are seeing.
As Azygous said day length has an effect. The big difference is whether the days are getting longer or shorter. If they are getting shorter that can trigger the molt and get them to stop laying and use the nutrients that were making eggs to grow feathers. As the days get longer that can trigger them to start laying. That affects different hens individually though. Some, especially the production breeds, may not wait for the days to get longer to start laying. But some will. They just have different triggers to start them laying again.
Another trigger is how they are eating. Before they start laying again they need to build up some reserves of nutrients. If the days are fairly short they may be sleeping more than they are eating so they are not getting enough nutrients for egg production. If you increase the protein levels you can help overcome that, or extend lights to give them more time to eat after they digest what they eat. Extending the lights can also make them think the days are getting longer.
A lot of people recommend you increase protein levels during the molt to help them get over the molt faster. Since the speed of the molt is mostly about how fast the feathers fall out feeding them extra protein isn’t going to speed that process up much. It will help a bit and the feathers will look really pretty when they grow back, so there is a little benefit there. But what is will do is enable them to build up that reserve of nutrients faster. Some individuals still have those day length or other triggers to start them laying again, but for a lot increasing the protein during the molt and just after can help get those nutrient reserves up and get them started laying again.