Mom went into the hospital 3 weeks ago as she couldn't breathe. 76 yrs old, COPD, on oxygen, no covid but positive for antibodies. Options?

Is home rehab with visiting nurses/therapists possible?
Hate to ask - if her cognition is poor, would she be able to tell she's in a SNF and not a hospital?

Sorry for your situation. Obviously, mom can not come home. Basically, the options are she has to leave the hospital and she has to go to rehab. Is Psych declaring her incompetent? If so, hopefully between you and the hospital staff, you can "make" her go to rehab. After that, she may very well end up in a nursing home, unless she does really well in rehab. Though if she is not mentally competent, memory care could be the proper place. The social worker at the hospital should be able to help you navigate this very challenging situation.

This not being able to visit people do to covid makes difficult situations soooo much more difficult.

If she has both Congestive Heat Failure and COPD she is likely getting very little oxygen to her brain. Sadly she is too ill now to confer POA on anyone and she will need guardianship and placement. If you wish to be guardian and place her in care do know that she is way too ill to care for in the home in all likelihood. The hospital will assist in transfer to rehab or SNF; she will need placement. A social worker can assist if you wish to act as guardian. If you do not wish to then allow the state to be her conservator and get her into placement where you can be the "good guy" and visitor instead of the "bad guy" that put her there. CHF means the heart pump is failiing. COPD means that the lungs cannot accept and exchange gases in the alveoli so she will have a lot of confusion ongoing and I doubt you would be able to do home care. She may soon need palliative or hospice care. I am so sorry. This is bad news.