Problem with my Uncle.

If grandma knew you were nosing around about her money, her heart would be broken in a million pieces. Is that the way you want her to remember you?

He can not restrict visits unless he has guardianship. Health care POA is a little trickier, depending on how it's written - but usually a POA is to act in a way that the principal would themselves - also many POAs are not legal or enforcable until the person has been deemed legally incompetent - in court, not just by a doctors letter. Many NH don't want to rock the boat or become intangled in a family feud so they'll go along with whomever is creating the biggest fuss. If you, your mom, aunt and your sister want to see your grandma and are ready to push the issue with the NH you are most likely within your legal right to do so.

The short answer is No, he can't do that, but it is a little more complicated than that. You mention he is financial POA, is he also healthcare POA? Because he is the nursing home's first contact they are apt to heed his directions, especially if he has been telling them you are a negative influence. What have you all done to get on his black list??
Your grandmother should meet with the admins herself at make it clear she wishes to remain in contact with all of you, if you could arrange a meeting with all the parties involved it would be even better.