You are correct to separate a single pair to a separate cage for breeding. A group of birds in a cage should never be given nests or nest boxes. So great move to start with! Are your budgies at least 2 years old? This is important because you want both birds to be fully mature and past any juvenile behaviors. It sounds like the pair is interested in breeding, and just needs more time. Preening each other and going into the nest box are good signs. Pairs need to feel safe and secure with their nesting area in order to get into breeding condition and mate & lay eggs. If the pair is close to the group cage, then you may need to move them to another room where it will be more quiet and private. Breeding birds do best when they have plenty of privacy. Especially a pair that you are trying to breed for the first time. A good diet is also important. Seed is not an adequate diet for breeding. They need a nutritionally balanced diet like pellets or our foraging diets, as well as dark leafy greens, chopped veggies and some fruit. And now that you have given them a nest box, you should start offering an egg food daily. This can be a dry commercial egg food or you can cook an egg with the shell washed, crushed and cooked with the egg. They should also have a cuttlebone.
It takes a lot of patience to breed birds, and you are likely to have some disappointments before you see any chicks. The first clutch almost always fails. Then you have to remove the nest box and rest the pair for 6 months. You must do this in between every clutch, limiting them to 2 clutches per year. This means you are likely looking at a minimum of 6 months to a year before you are likely to see any results. And if they do produce chicks, you need to be able to find new homes for the chicks, because they can’t live with the parents, and opposite sex chicks can’t stay together. You should never allow related birds to breed, so keep this in mind. The young chicks can’t be added to an adult flock or they are likely to get bullied and attacked by the older birds. So it’s a lot of information to take in, but just be patient and do the best you can on your end so hopefully the birds will eventually do their part.
Thank you for asking Lafeber,