It's easier if your budgies are trained enough that you can easily get them back into the cage.
All taming and bonding takes a great deal of time and patience.
To build your birdsí trust, sit by their cage and read, talk or sing quietly to them for a period of at least 10-15 minutes, 3 or 4 times day. After about a week, rest your hand on the outside of the cage when you talk to them so they will learn that your hand is safe and will not hurt them.
After a week of resting your hand on the outside of the cage, rest your hand inside the cage when you talk.
Donít make sudden moves, donít try to touch them.
Let their get used to the idea that the hand is now in their safe place and not harming them.
After 2 weeks, begin moving your hand slowly toward your bird. If they become agitated, stop moving your hand and just hold very still until they calm down. When they are comfortable with your hand near them, you can offer them a bit of millet or a few seeds.
Always work at your birdsí pace.
Move slowly and talk reassuringly and calmly to their whenever you interact with them.
Be sure to bird proof your bedroom before allowing the budgies out of the cage.
You can tape sheets or netting up a few inches from the walls to ensure the budgies don't fly into the walls until they get used to the room's dimensions.
I have untamed budgies that go back in their cage because I've used positive reinforcement training to teach them they will get a reward when they do so.
Initially, when I was ready for them to go back in the cage I would first dim the lights and pull down the shades in the room. If the radio or TV was on, I turned it off.
Then I would stand by their cage and ringing the bell on one of their toys while telling them, "It's time to go in your cage now!"
Once they went back into the cage, they were given a small bit of millet.
This method worked well and they now go in the cage when asked to do so without the lights being dimmed or the blinds lowered and the TV or radio can stay on.