Okay, first of all the eggs probably are not fertile if you had them originally separated, so it's likely that this clutch won't hatch...Just them being together preening and such probably causes the female to become hormonal, which causes her to lay infertile eggs. So I wouldn't be surprised if this clutch isn't fertile, but you can "candle" them in about a week (after each egg is a week old) by holding a flashlight behind them, and if you see any red veins then they are fertile, if not then they are infertile...
Are you saying that you want to remove the babies from the nest-box and hand-raise/hand-feed them yourself? Because if you've never done this then I don't suggest it at all, I suggest you leave the eggs/babies in the nest-box full-time, do not move them, and allow the parents to feed them and raise them, wean them, etc. If you want them to be "hand-tame" then all you need to do is wait until they are 3 weeks old and then block-off the nest-box entrance when both parents are out feeding with a piece of cardboard, and then spend 15-20 minutes each day just sitting with/handling each baby (make sure you don't do this until they are 3 weeks old because they are so small, you can easily kill them)...If you have no hand-feeding experience then you will most-likely only end-up having them not make it if you decide to remove the chicks from the nest-box and hand-raise them...
What do you have a "lamp" for? Babies in a nest-box being raised by the parents need no artificial light at all, and you'd kill them from heat-stroke if you put a lamp over their nest-box. The parents take turns laying on the with Budgies, and they are fine. You don't have to do anything at all if you allow the parents to raise them except provide the parents with plenty of food and water...
And if you were to remove the babies to hand-raise/hand-feed them yourself (which you do no sooner than 2 weeks old and no older than 3 weeks old if you do, which I don't think you should at all), you must have a Brooder to keep them in, you cannot put any type of "lamp" over them!!!! It will not keep them warm enough firstly (before they get their down they must be kept at 95 degrees F, and then after they get their down but before all of their mature feathers grow-in they must be kept at 85 degrees), but also YOU'LL END-UP BLINDING THEM IF YOU PUT A LAMP OVER THEM!
It's not optional to not have a proper Brooder, or at least a homemade one that has a thermometer in it and is set-up to keep the babies at constantly the correct temperatures, otherwise they'll either die of exposure, or more-commonly they'll die of severe fungal infections in their Crops/GI Tracts because if they are even a degree too cold they cannot properly digest their formula...The same goes with the hand-feeding formula that you would have to feed them every 2-hours for the first 3-4 weeks (including every 2-hours overnights), you have to always have a candy/cooking thermometer in the hand-feeding formula you are feeding them, as it must ALWAYS be between 104 degrees F and 110 degrees F...1 degree cooler they will develop Fungal Infections, 1 degree hotter and it will burn their crops and cause a horrible wound and infection...So you must have all of the necessary equipment, proper commercial baby bird hand-feeding formula, etc. to remove them from the nest-box and hand-raise them...And please, no lamps!!!
If you want them to "raise them naturally", which is the best idea by far, then you don't need to do anything at all except provide the parents with lots of food and water. The egg-food is a very good idea, as is a Cuttlebone and a Mineral Block...but that's all you have to do. Don't "remove the babies" from the nest-box at all except to start handling them for 15 minutes a day after they are 3 weeks old. That's all you need to do. No extra heat sources, no "removing the babies at 4 weeks", etc. The parents will do it all.
