Yesterday I introduced Blue, a rescued parakeet, to my two English budgies, Beau and Happy. Although the comparatively calm Beau and Happy were skittish about Blue's hyperactivity, they all napped together and Beau even played briefly with Blue. At night Beau and Blue slept on the same perch and there was no activity in the cage for the first six hours, after which I went to bed.
This morning, I found Beau on a rarely used perch near the bottom of the cage. When I opened the door, he immediately climbed up to my shoulder and stayed there for an hour. Since then, Beau has either sat on my shoulder or perched on the playground. He won't go back in the cage, and takes flight whenever Blue approaches him -- which isn't constant, but frequent enough to be very disruptive to boss-bird Beau's routine.
Blue has been neither aggressive nor territorial. He seems to just want to be where the other birds are, doing what they're doing. But Beau remains frightened, even to the point of not wanting millet (which all three shared yesterday). Happy, meanwhile, is not as scared, but jumps to a different perch whenever Blue approaches him.
I would hate to have to put Blue back in his quarantine cage, because he really wants to be with the other birds. But I'm thinking I may have to do that, at least temporarily, and hope Beau will become less afraid over time.
It is important to go slowly when introducing new budgies into a current "flock".
Did you put the two cages side by side for a few days to allow the birds to become accustomed to one another through the cage bars?
Did you then introduce the three birds in neutral territory several different times before allowing Blue into Beau and Happy's cage?
I would suggest you back up, place the two cages next to one another and allow the three to enjoy out-of-cage time together for a few days.
You can leave the cages open and let the budgies decide which cage they want to be in and who they are most comfortable with.
When you have an odd number of budgies, one often gets "left out" ends up as the third wheel or may get picked on by the other two birds.
Yeah, I placed the two cages next to each other and thought all was fine because the birds' behavior didn't strike me as that different from what I observed twice before when I introduced a third bird. But different birds, different personalities!