Depends on the setting; In long term care, quite a lot. (People are very incontinent all the time, but the human in attendance of the robot still has to clean and care for the patient, while the robot takes care of trash and linens. Most of the time spent (more than 70%, upward of 90 if the patient is uncomfortable and needs reassurance) is with hands-on patient care for those doing grunt work. Patients with dementia may be obstinate about refusing care, but still most definitely require it; the care giver has to convince such patients to submit to receiving the care, and that process can be quite lengthy. This is why you would need a strong general AI straight of of science fiction to replace nursing staff. (Office and desk nurses could probably be completely automated, if not for the requirement of needing quality communication with doctors. See also "Call center robot hell, hospital version" and now people die from it.)
"Nursing staff" covers CNA work (which mostly focuses on pericare, dealing with belligerent patients, and the gruntwork of getting room trays and delivering specimens. The latter two being ad-hoc errands that are infrequent but annoying, the former being the mainstay of occupation) and RN work (Foley insertions, wound dressing and care, setup and operation of IV units, Skin and wound assessments, detailed medical documentation collection and communication, and management of CNA workers). Since there is a shortage of CNAs, RNs often do CNA duties too. (It is not uncommon to see an RN cleaning up puke and shit.)
Robotic assistants could cover a portion of CNA work, but very little of RN work.
For acute trauma care, even less. RNs do much more because there is more that needs their higher credentials. Working a trauma ward has stricter requirements, and is harder on the mind.
For psychiatric facility care, nurses tend to need CNAs more, and "human interaction" is needed more by the patients. (A robot is likely to be destroyed by the patient in a fit of rage, since the patient is likely to be literally psychotic in this setting.)
For something like "assisted living", where the patient is rarely incontinent, and just needs help remembering to take their pills, put on clean clothes, and eat regularly-- Robots could probably do a whole lot. Assisted living is a vanishingly small demographic though. This is the demographic that the Japanese are targeting.