How can I stop my brother moving my mum to a nursing home far away from me?

Can you give us more details?
Why is your brother moving your Mom?

You probably can't, unless there is a clear risk to your mother's wellbeing that would cause your local social care authorities to intervene.

Your mother might be able to, though, if she doesn't want to move. Would you like to describe what is happening?

Maybe go into some detail. The way you're stating this is kind of ambiguous. Where do you live? Does mom live near you? Come back with more information and you will receive more help.