If the babies are only 1-2 weeks old then they are being fed every 2 hours or so, you need to make sure that both your adult Budgies have access to plenty of seed-mix and fresh, dark leafy greens and fresh veggies ALL DAY LONG! Both the mother and father Budgie will be feeding the babies, and especially the mother, because she also made and laid the eggs, tend to lose weight, calcium, vitamins, etc. at an alarming rate while feeding babies, so they both must always have a large dish of a healthy seed-mix in their cage at all times. You don't need to put it in the nest-box, they'll both take turns sitting on the babies and eating, so just make sure there is always a large dish of seeds in their cage at all times, along with lots of dark, leafy greens and veggies.
You also must have both a Cuddlebone and a Mineral Block hanging in their cage at all times, so that they are getting plenty of Calcium, especially the mother. She'll likely start laying eggs again pretty quickly after she finishes weaning these babies, so she needs plenty of Calcium to prevent egg-binding. You can buy a Cuddlebone at pretty much any Walmart, who also sells Mineral Blocks too, or any Petco/Petsmart. And buying a bag of Qwiko Egg-Food at Petco is a good idea too, and making sure there is a small dish of that in the cage for the parents every day until they are done feeding the babies is a good idea.
As far as handling the babies, once they hit 3 weeks old you can start blocking off the nest-box when you see both parents out of it, use a piece of cardboard or something similar once you see them both out of the nest-box, so that they can't get back in while you're handling the babies. At 3 weeks you must be extremely careful and gentle, and only handle them once or twice a day each for 15 minutes or so. And then each week you increase the time you handle them/hold them/pet them, but you can't keep them outside of the nest-box for "an hour" until they are fully-feathered with their mature feathers, not just their down. Without their mature feathers they cannot regulate their body-temperatures, and that won't happen until they are around 6 weeks old. At that point they will also be starting to wean and also be ready to fledge, so a good rule to follow is once you see the babies starting to venture out of the nest-box on their own, then you can take them out for a longer time period. But in-order to hand-tame them you do want to start just holding them, petting them, and talking to them once daily, every single day, for 15 minutes each. Again, be very careful as they are going to be very fragile at 3 weeks old...Best to sit in a chair while holding them because if you drop them they'll likely die of internal bleeding.
You may also want to think about weaning the babies onto a healthy, natural-flavored pellet instead of just a seed-mix, because it's difficult for people to switch Budgies over to pellets if they are weaned onto seed-mix. Buy placing a big bowl of a healthy, natural flavored pellets (no fruit flavored pellets, too much sugar) inside of the cage when they start weaning you'll encourage them to wean onto them early, and of course you must also make sure they have access to fresh veggies while weaning too. Budgies usually start weaning around 5-6 weeks old, and are usually fully weaned between 8-10 weeks old.
When the babies are fully weaned and completely out of the nest-box, remove it immediately. It's likely you'll find an egg in the bottom of the cage or in a food dish soon after the babies wean. If you do, do NOT put the nest-box back in the cage! Instead, you need to remove the egg immediately when you see it, boil it on the stove for 20 minutes, let it cool, and then put it right back in the cage on the bottom grate. You'll need to do this immediately to each new egg that she lays, so pay close attention checking the bottom of the cage and the food dishes twice a day, once in the morning and then again at night. She won't start laying on the eggs in the bottom of the cage until she lays at least 2 or 3 of them, but make sure that you are boiling each individual egg as soon as you see it laid and then just placing each one back on the bottom of the cage together (NO NESTING MATERIAL/BEDDING AT ALL!)...just lay the boiled eggs right directly on the grate on the bottom of the cage. Once the female realizes that they aren't going to hatch she'll stop laying on them, and then you can just remove them and throw them away once you're sure she's no longer laying on them. Hopefully this will knock her out of breeding-season and stop the egg-laying. Never put a nest-box or any type of nesting material/bedding/wood chips/ boxes/"huts"/tents/blankets or towels, etc. in their cage again, as all of these encourage their hormones and she'll only start laying eggs again...And just be sure for the future to simply boil any eggs you find in the cage and then place them on the bottom for her to lay on and eventually get bored with, then pitch them.
Cockatiel
Cockatiel
Cockatiel
Budgie
Lovebird
Green cheek
B&G macaw
some start trying at 7-8 months, so be prepared...)
