I've been caring for my mother for years in my home, but just because she tripped and hurt her foot my brother and sisters put her in a nursing home and said I'm incapable of caring for her. She tells me every day that she wants to come home. My family is going on vacation and don't want to be bothered. I want her home with me and need to know if legally I can bring her home without their consent.
If your mother does not have mental capacity, then you as her daughter can still arrange for her discharge from the Nursing Home; but that doesn't mean it would be a good idea, necessarily. It would be much better to get the formal legalities sorted out first, ideally applying for guardianship, and you should take advice from a lawyer who specialises in Elder Care; or at the very least discuss all the issues with an experienced social worker - the Nursing Home should be able to suggest someone for you to talk to.
But you're not so concerned about your family's opinion on this for no reason, are you? Can you perhaps tell us a bit more about how come they seem to have taken charge, and you have let them?
I have a feeling your siblings felt that since Alzheimer's/Dementia only gets worse, they did you a favor by placing Mom in a Nursing Home. Before you know it, you are caregiving 168 hours a week, getting 2 to 3 hours of sleep at night, to a point of exhaustion, crashing and burning, then what?
Oh, by the way, when someone has Alzheimer's/Dementia and they say they want "to go home" they mean their childhood home, where life was simple and fun.
I have a feeling one of your siblings has Power of Attorney in order to sign admission for your Mom to live in a Nursing Home.