Hi Henry and
to the forums!
There's no reason to be scared for your birds; as long as they are eating, drinking, and playing normally, there's no reason for concern whatsoever.
Budgies naturally take a long time to adjust to their new environments, and after four months, they are most likely comfortable with each other and their cage but not with the outside of their cage.
This is perfectly normal.
Have you worked with your budgies yet with trying to get them comfortable with you? Budgies who trust and relate to you are more likely to want to come out of their cage to spend time with you.
Also, as mentioned above by Therm, it's important to have a reason for them to come out-- for example, you can put the play gym next to their cage where they can see it, perhaps put a perch on the outside of the cage by the door, or put a favourite toy outside the cage by the door so they can't quite reach it unless they come out.
Additionally, be sure that the room looks and feels safe-- no other pets present, all windows and doors closed, and glass covered to be sure that the budgies don't feel threatened. Playing soft music for them helps, too. If you simply take a computer or a book in there and sit near to them while their cage door is open (and all other doors are closed), they may start to get more comfortable with exploring the outside world.
Remember, this all takes time, so try not to think something is wrong with your budgies simply because they don't want to come out instantly. Their cage is their "safe space", so it's natural for them to want to stay in, especially if they're happy and comfortable.
I look forward to meeting your budgies when you get a chance!
It's great to have you with us, and you couldn't have come to a better place to learn even more about the best of budgie care!
Be sure to read through the forum's Budgie Articles and "stickies" (threads "stuck" to the top of each subforum for easy reference) to stay up to date on all aspects of budgie care. If you have any questions after reading through everything, be sure to ask as we'd love to help!
We look forward to seeing you around the forums