They are 20 weeks old so they could start laying at any time. Personally I’d leave the nests open so that first controlled egg can be laid in a nest. When pullets start to lay, some just drop an egg wherever they happen to be, the roosts, the coop floor, or out in the run. Most have control over this right from the start but many don’t. So don’t be too surprised to find eggs about anywhere. When a pullet does lay that first controlled egg, she looks for a good place for a nest. And once she lays that first controlled egg where she thinks is a good place she instinctively wants to go back to that spot for all her following eggs. Her instinct is to build a clutch of eggs even if she never goes broody. This can be a hard habit to break.
Pullets and even older hens also tend to want to lay where another hen is laying. They seem to think that makes it a good place to lay if another hen is laying there. That’s where putting fake eggs in the nest can really help. I use golf balls but there are plenty of alternatives.
Another reason I want the nests open before they start to lay is that if there is a problem I want to know about it before they start to lay so I can fix the problem. Often about a week before she starts to lay a pullet will start investigating possible nest spots. Sometimes a rooster will help her in her search. They will often scratch around to see how good a nest can be made. If the bedding or fake eggs are scratched out of the nest, you have a sign that something is wrong. Usually that means the lip is not high enough to keep things in, so raise the lip a few inches.
Also, you some may sleep in the nests. I want to know that so I can fix the problem before I start getting poopy eggs. You’ve got that problem identified in time to fix it.
One probably cause from your post is that the roosts need to be noticeably higher than the nests or anything else you don’t want the chickens to sleep on or in. Normally 12” higher is noticeable, even if it is across the coop. So the first thing you need to do is to raise the roosts or lower the nests so there is a noticeable difference.
Do you have Silkies or some other kind of chicken that can’t fly? Those can be special cases, but the vast majority of chickens will have absolutely no trouble getting up on a roost 4 to 5 feet off the ground at 20 weeks. I’ve seen two week old chicks fly up 3 feet to roost when the broody hen tells them “Children, come on up here. NOW!”. Since yours are getting to the nests they don’t seem to be having any problems getting up there but if you are concerned about that, put in an intermediate perch about half way up. They might use it, either to go up or come down, though mine normally ignore any perches on the way down and just fly to the ground.
There is one warning I need to give. Chickens spread their wings, both to go up and to come down. They need enough clear space in the coop to spread those wings. Nests, feeders, or waterers in the way can create problems. Normally the higher the roosts the more clear space they need to come down especially. I recommend you keep the roosts as low as you reasonably can while making sure they are noticeably higher than the nests.
Since chickens are creatures of habit, yours are probably already in the habit of sleeping in the nests. Just making sure there is a noticeable difference in height will probably not be enough to break that habit. I know of two ways to do that. One is to go out after they are asleep and the coop is dark and physically take them from the nest and set them on the roosts. The coop needs to be dark so they have trouble getting back to the nest if they want to. Use as little light as you can. They should get the message fairly quickly, though some can be more stubborn than others.
Another method that is a little harder is to block the nests off after the time for laying is over for the day so they can’t roost in the nests. You may need to still put them on the roosts to show them where a safe place to roost is. Then you need to be out there before it is time for them to lay to open the nests back up so they can lay that first egg in the nests. So you need three trips to the coop; 1) to block the nests before they go to bed, 2) to make sure they are on the roosts instead of somewhere else you don’t want them, and 3) early the next morning to unblock the nests.
Or you can block the nests all the time, possibly teaching them to lay somewhere else, then start a new thread on here on how to retrain them to lay in the nests instead of somewhere else. Another reasonable approach since they are not laying yet is to block the nests until they get in the habit of sleeping on the roosts and then opening the nests back up. Maybe you can do that before they start to lay.
As you can probably tell we often have different opinions on how to do things. I know I’m being a little snarky in the last paragraph, but there are usually a lot of different ways to go about these things. Usually it is not a case of just one right way to do these things and every other way is wrong, but more of a case where there are many different things that work. You just need to pick one that seems to fit your situation. But get out there and make sure there is a difference in the height of the roosts and the nests. That is important.
Good luck!