I don't know if this would help but have you asked her why she thinks your budgie would spread diseases? And what sort of diseases is she referring to? My mum threw the same thing at me and did not want Preeno in the house because she thinks birds are unsanitary.
I don't know if there are articles that show that birds are in fact very clean creatures. If your bird isn't kept in unhygienic or overcrowded conditions and is healthy, I don't think there is any reason to worry about it carrying diseases. Perhaps dander and dried poop could be an issue but that is easily fixed.
It's hard to give advice not knowing what your mum is like, but perhaps you could 'bargain' a little - ask for 1.5h and show that you can have your budgie out responsibly, especially if cleanliness is one if her concerns. Then maybe slowly increase the amount of time.
Sorry if this doesn't give you an immediate solution. I know how difficult it is and hope you manage to convince her!
She think that's swine flu and other diseases that spread from birds will turn us sick, and even now I take them our for 1.5 hrs (it had first been 30 minutes) , and the room always stays completely clean (except for some poop but I always clean up after them) but she isn't that concerned about dander or dried poop, but diseases that spread through contact, and she thinks that if Skype and Twitter fly around freely in my room, they'll infect the whole place. And they both stay in a clean and hygienic cage, there's no way they have diseases they can spread but she apparently "doesn't want to take the risk"
And I'm trying everyday. I showed her some articles that state that budgies very rarely spread diseases (less than 10 people in 2010) but she always says, 'don't push me, or I'll give them back to the pet store.'
Maybe if you guys post here that budgies don't spread diseases and are quite sanitary birds, I'll show this thread to her and she'll agree?
Oh, also did your mom let Preeno out afterwards?
First thing I'd do is start a savings for a larger cage.
Your mom may never change her idea, and it's better for your bird if he has to stay caged.
If you mom likes scientific facts, prepare a report on the amount of space and area a wild budgie utilizes on a daily basis.
How far they fly, the amount of searching for food, etc...
Make sure you prove responsible ownership by keeping the cage spotless, and doing other things mom likes.
Earn the right to ask for more time with your bird.
It's up to you to provide the best life for you bird, no matter if caged or not.
Devise entertaining toys, foraging in the cage so your bird has things to do.
It's not a great situation, but you must do what you can to make the best of it, and hopefully in time win (not argue) your mom over.
I'm just guessing that 'diseases' is an easy excuse to keep the bird mess to a minimum.
Once she sees that no one is catching these bird illnesses, that excuse will fade over time.
Earn her trust in your dedication to proper care and behavior as a bird owner.
I actually take care of everything; changing food, water, cleaning cage, taming them, but her main argument is that she'll let me do everything I want with them as long as they're in their cage, not out. It took my days of wheedling to finally get her to allow them out for an hour.