I've done this on an old wool blanket and it worked, but i don't know if it will work on an old teddy bear because i don't know what that fabric is. Cotton? You can try lemon juice and salt together (an abrasive and an acid) or perhaps vinegar and baking soda (haven't tried this, but maybe it's gentler than lemon juice?) but go really really gently on a tiny area with a very soft toothbrush to see if that works.
Then I saw this: (maybe try this one first?)
Allow your wool item to sit in the sun and dry. The mildew will dry out also and turn into a powder. Then, brush or wipe off the mildew powder onto some old newspaper or into a bag.
Do you want to put an old teddy bear in the sun though? I'm not sure.
Good luck! I'm wondering what other people will say. Let us know what happens please. :0)