hi, i have a large breeding cage. do you think i can hold 1 female budgie and 2 males for her to pick to breed with

I would strongly advise against putting 2 males and 1 female together. Its only going to cause terrible fights between the two males, and sometimes they will fight to the death!! So choose which male you want to pair up with the female and keep them in a separate breeding cage on their own. If theyare in breeding condition they will pair up eventually. Be patient and feed lots of vegies, cuttlefish,good quality seed and greens. Good luck.

how long does it usually take for budgies to pair up and start to breed when you introduce to a nestbox?

It all depends. I have had pairs start breeding within a few weeks. But then I've had some that have taken a few months to show any interest. Make sure you dont breed them more than twice a year as it takes a lot out of the hen to rear her chicks. So you need to give them a break so they can recover and build up her calcium supplies again. Her cere (the brown part above her beak) should be a dark brown colour and the male cere should be a nice deep blue when ready to breed.