George, son of Peter I of Serbia (1903-21) was declared schizophrenic and passed over in favor of his brother, Alexander I. Evidence for his schizophrenia derives from his own memoirs.

Don Carlos, son of Philip II of Spain (1556-98), was disinherited and confined on his father’s orders, dying some six months later. His ailments appear to have been both physical and mental in nature -- possibly the result of brain damage incurred at birth, but later exacerbated by a head injury.

Ludwig II of Bavaria was pronounced to be "in a very advanced state of insanity" and "quite incapable of exercising government." After drowning shortly thereafter under mysterious circumstances, he was succeeded by Otto I, who was quite deranged and kept under restraint ever since a schizophrenic breakdown in 1871. He reigned until 1913, when he was deposed.

The first two monarchs of the Scottish House of Stewart, Richard II and Richard III, were both considered unfit to reign for reasons of instability.

The Russian Tzar Ivan IV ("Ivan the Terrible") is commonly accepted to have suffered from a pretty serious mental disability, expressing itself in bouts of violent outrage. During just such an outburst, he killed his son and heir apparent, leaving the succession to be passed to Ivan's younger son, Feodor I, who was feeble-minded (Check out the painting by Illya Repin "Ivan the Terrible and His Son" for a haunting rendition of this scene, as well as for the look of madness captured in Ivan's eyes). Feodor, though technically the Tzar, was acknowledged as too disabled for rule and so a regent took his place, while Feodor himself in said to have roamed the lands ringing church bells.

Sources:

https://www.academia.edu/7766998/Two_Feeble_Monarchs

The Greenwood Encyclopedia of International Relations, Cathel Nolan

The fourth Roman emperor Claudius suffered a limp and from a speech impediment that caused him to stutter and was widely mocked by certain members of the imperial family and the aristocracy, including his own mother, and his grandmother Livia.

Because of his frail constitution, Claudius spent his time indoors reading, often interacting with the great historians of the time. He gained an interest for history and was allegedly the last person to read the Etruscan language. His two best know works were a history of the Etruscans and a History of Carthage, but they are lost to us now.

Claudius survived to middle age perhaps due to his disabilities, he was not threatening and therefore there was no reason for other members of the imperial family to assassinate him, though Caligula in his madness did come close a few times.

Eventually, during a coup against Caligula by the praetorian (the first time the praetorian would assassinate an emperor), the praetorian began to fear reprisal by the ex-emperor's Germanic body guard. To counter this, they proclaimed Claudius emperor, as they figured that the Germanic bodyguard would be appeased by this fact . The ploy worked and the Germanic bodyguards did not attack the Praetorians.

Claudius would reign for 14 years during which time he would build the city of Rome a new port to supplement and eventually replace Ostia as the main grain port. He would also conquer Britain and used elephants during the invasion (Julius Caesar had invaded the island a hundred years earlier but had failed to hold it). Claudius would eventually die, possibly from poison and be succeeded by Nero.

So Claudius was hardly a fool, but survived through a comedy of errors and good fourtune to become emperor of Rome and managed to improve the infrastructure and enlarge the territory of the empire.

This is not exactly what you're looking for but there is a book called Mad Princes in Renaissance Germany that describes a few different mental illnesses among royalty and how they were dealt with... mostly in the 1300-1500s.

There are a few cases where older rulers suffering from "melancholy" which is probably often dementia or schizophrenic disorders being removed to be replaced by a family member. The best example of this is probably Friedrich of Brandenburg-Ansbach whose sons basically imprisoned him during a sort of "Opposite Day" carnival in order to take power.

There are also examples of younger rulers like Duke Wilhelm "going mad" and shooting his pistols off walking through town and yelling into the dark. His case wasn't wrapped up as easily and even when he was mutilating himself and his council with a knife the sternest treatment he received was his father writing to him urging him to stop.

There are all sorts of interesting removals of royal members due to illness in this book. Another good one is a young princess who is widowed and is spurned by other young princes at her old age and is basically institutionalized in an old castle that housed widows after she goes insane. There are even instances of "madness" brought on by physical trauma like a prince who is injured in a jousting tournament which was probably a severe concussion.

To what extent are mental disorders like schizophrenia constant throughout time? At least the diagnosis of some of them, if not the way people with them behaved must be dependent on the cultural context no?

P.S. Besides this, George III of the UK (r.1760-1820) is famous for having some mental disorder (porphyria has been suggested, see: Peters, Timothy J.; Wilkinson, D. (2010), "King George III and porphyria: a clinical re-examination of the historical evidence", History of Psychiatry 21 (1): 3–19) - he had bouts of 'insanity', with different 'episodes' including 1788-9, and eventually in 1811 his son took over as Regent with full monarchical powers until George's death in 1820.