How long until my budgie is comfortable with my finger in its cage?

It may take time and patience. You can begin by simply sitting next to the cage for short periods of time initially and increasing it as they acclimate. They will flutter about and move to the far side of the cage at first. Eventually, they will settle down and accept your presence. Move slowly and speak softly around the cage. Once they have completely acclimated, you can progress to placing your hand in the cage. Make no attempt to touch or hold them. Again, they will flutter, panic and move to the far side of the cage. Leave your hand inside the cage for ever increasing periods of time. Use small treats as rewards after each session. Eventually, they will become curious and may hop onto your hand if you have treats. You can teach the 'step up' command by extending a finger and gently pressing it against the chest while saying "Perch" or "Step up". From this point, you can work with them to teach additional tricks or whistles.

Patience is best when getting your budgie used to handling. If he has a favorite treat you can offer it every time you approach the cage or every time he sits up on your finger. Once he is comfortable there, you can begin to then offer treats after you pet him, to teach him that this is a rewarding behavior. It can take some time, and if your budgie is too stressed simply take a break and let him calm down a bit before trying again.

It may take a little longer than you thought to get him used to you. Some budgies, if they weren't hand tamed before you got them as little babies, they may always be a little skittish. Try putting his favorite snacks and foods on your hand and holding very still and see if comes over to nibble at that. Keep talking to him and singing to him softly all day so he gets used to you. Try sitting right next to him without moving as well.

Time and patience! It can take several months for a bird to start trusting enough to want to hop on you or interact more. It sounds like you're on the right track already with offering food from your hand. I would continue to do this, and gradually increase to petting Sky, and then handling her more/asking her to perch. If she becomes too nervous, letting her calm down and trying again later is best.

The squawking and knocking stuff over is normal, so I wouldn't worry too much about that. It's best to get bowls that can be fixed into position to prevent this. It generally takes them 2-3 weeks to start getting comfortable in a new environment. It's definitely good to interact with them and get close to get them (putting your arm in the cage is good) more familiar with you. Offering treats is always a good way to make friends. Eventually they will come up to your hand and may perch there, but don't be surprised if it takes a month or so. I hope that helps and all is well with your new Budgies.

Give Buddy some time! It takes awhile for birds to get used to new surroundings. Make sure he has many perches, toys, and fresh food and water available, and try to leave him alone for a few days to adjust to his new environment.

It will take time. Allow him to keep eating from your hand. Use treats to coax him onto your hand and when you pet him, do not come in from above. This can be a threatening motion for birds. Approach at or below eye level and move slowly. The female will require a bit more work in getting acclimated to your hand. You can do the same though. Use treats and rewards. Keep your hand near her, but don't try to touch her, for ever increasing periods of time. Provide additional food and water bowls to prevent resource protection.