How do I move loved one from one nursing home facility to a facility closer to me?

I would start by going to the facilities near to you and asking if they can accept your father with his particular needs. It is important to know they will ask you about assets, so before going you and your mom should consider going to a good CELA (certified Elder Law Attorney) for an hour or so of advice about division of assets for care so that your mother's assets are not eaten up with care for your father.

Your father would at some point qualify for the help of Medicaid, then, while some of your mom's assets were preserved for her own care.
You will need to decide who is to serve as POA for your father if he is competent enough to appoint one. If there is already a will a Springing POA would likely be in it appointing your mother. She could serve with your helping her, but record keeping is needed.

I would advise you to start a diary now of everything to do with your Dad on a day to day basis.
I sure do wish you good luck and am glad your mom has your support here.

I agree with Geaton777, it is "hospital delirium". This can happen to anyone of any age, even a teen-ager after having major surgery. Doubt a nursing home would say a teen had dementia, even with the same type of confusion as an elder.

Was he in the hospital before going to rehab/NH? If so, he may have a temporary condition known as "hospital delirium". If so, it should resolve on its own over time. But there are other factors that can contribute to it: a UTI, dehydration (if your Dad is not drinking or eating enough in the NH they won't force him or encourage him to do so), anesthesia; certain antibiotics, any other pre-fall health issues, etc.

I'm not sure he should be "diagnosed" with dementia until he's been give some time to come back around. And he may not, but you won't know until some time has passed.

Jan 2023 my 100-yr old Aunt with adv dementia fell in her home and broke her hip. She went to a reputable rehab clinic because she wasn't going to have any surgery to repair it, but they were trying to get her to the point of pivoting on it. Our family then realized that this is not the same as a NH or SNF. They would put her food tray in front of her and then assume she would feed herself, but she wasn't. Our family members had to do it because she was so out of sort in her new environment.

I'm just saying do not assume what the staff is able and willing to do for your Dad and don't assume it in a different facility. They cannot force him to do anything, including drink and eat. They can't force him to keep a fluids IV in. etc.

If you have the option of a family member going there to make sure he is getting/doing the basics regarding eating and drinking, it may help him get over the delirium. UTIs are extremely common in the elderly because of the changes in their physiology, and if your Dad had a catheter in at any point, increases the risk.

Read this:

https://www.hopkinsmedicine.org/health/conditions-and-diseases/delirium#:~:text=Delirium%20is%20common%2C%20showing%20up,awareness%2C%20hallucinations%20and%20mood%20changes.