Ok, all my birds are now settled and covered for the night so I can post photos now.
My Budgies are taken out to their aviary every morning and then brought inside every night. The last time I checked on them was 5pm and they were all fine. When I went outside at 9pm to catch them up I saw Bubblegum clinging on the wire and she had a massive red lump protruding out her vent.
I caught her, applied KY Jelly to the area to keep it moist, and then rushed to the emergency vet. There was horrible traffic on the highway because of roadworks and idiot drivers so the 20 minute journey actually turned into a 40 minute one!
Got there and the receptionist took her to the vets in the back. I've been lucky enough to do some work experience at this emergency vet clinic so I knew Bubblegum was in the best possible hands at this time of night.
About 25 minutes later the vet brought Bubblegum out to me. They massaged a stuck egg out of the prolapsed cloaca and then pushed the cloaca back in. I could now take her home.
She is a bit shocky so she is covered in a dark, quiet, warm room. She has blood EVERYWHERE but I'm not going to catch her now to clean her up. We'll be going to our avian vet first thing tomorrow so they should be able to clean her up for me.
I've also got some Clavulox to give to her but I'm going to wait until morning because I feel it is too risky to forcce medication into her beak at the moment.
Here's the egg:
Bleeding has completely stopped, but it was dripping from her vent right after the procedure:
And her backside is a mess, but it's all cosmetic at the moment:
Sorry for the quality of the photos.
EDIT: I just wanted to add that for members in the Brisbane area with an after-hours bird emergency - you can't go wrong with Animal Emergency Services. Even though they don't have an avian specialist on call, they have ALWAYS been able to help my birds and have them pull through until my regular avian vet could be contacted. Don't hesitate to take your birds there in an emergency just because they usually only see cats and dogs.