Three months is not very much time when you are trying to get birds to breed. It takes a lot of patience to breed birds, and involves a lot of work, time, expense and often disappointments on your end. You were definitely right to remove the extra male. You can’t breed birds with extra birds in the same cage – you should only have one pair of birds per cage. American budgies have been colony bred, but it is something done by for profit breeders and you have no control over which birds breed and whether inbreeding happens(related birds breeding).
As to your pair, they may just need more time to get settled in. This may be months or over a year. They need a lot of privacy so that they feel they have a secure nesting place. The nest box should be affixed to the outside of the cage, as high as possible. I would not give up on offering cooked eggs, with the shell washed, crushed and cooked with them. The extra protein and calcium can help trigger egg laying and the female needs extra protein and calcium when she is nesting. They also need longer days – longer exposure to light – and warm temperatures in order to breed. If they do mate and lay eggs, the first clutch will almost always fail due to mistakes made by the parents. But they need to learn in order to be good breeders. Bird breeding is not easy at all. You can have everything set up exactly perfect and still see no results. Not all bird will make good breeders. A bird might be too nervous, infertile, too young or too old or many other causes that prevent production. All you can do is provide what they need, leave them alone, and hope they do breed.
Thank you for asking Lafeber,