Austira. Archduke Carl ludwig the fourth son of Em,peror Karl was born in march during the reign of his father.
As for württemberg duke Carl Eugen became Duke at a very young age and had one daughter by his amrriage to Elisabeth friederike of btrandenburg-Bayreuth. Beofre him also Duke Eberhard Ludwig became duke at a very young age and his only son was therefore born during his reign.
I believe the twin children of Prince Albert II of Monaco and his consort, Princess Charlene, were the last European royals born to a reigning monarch. Certainly the spotlight is high, when an enthroned sovereign becomes a parent -- especially when a direct heir to the throne is born. Before, he and his own siblings were likewise all born to a reigning monarch, his own father (Prince Rainier III).
It would be too exhausting to run through the entire list; and certainly in the past, before modern medicine (when death rates were high and life expectancies low), it was a common occurrence for a young (sometimes even minor) royal to succeed to the throne as sovereign. In that case, one would have had a reigning monarch marry and have children. So I thought of going through more recent history (relatively speaking), in both reigning and non-reigning houses.
Monaco: when was the last time this happened, before the children born to Prince Rainier III (who succeeded young because his mother, Princess Charlotte, renounced her rights in his favor)?
Liechtenstein: but for the renunciation of rights of Prince Aloys (1869-1955) in favor of his son, Prince Franz Joseph II would not have succeeded to the throne young, as a bachelor. As it was, he married in 1943, five years after his accession: so all his children were born to an enthroned sovereign. How about before?
Luxembourg: all six children of the Grand Duchess Charlotte were born to a reigning monarch, since she married in 1919, later in the same year she succeeded her older sister (Grand Duchess Marie Adelaide) as sovereign. Before, one could perhaps count Crown Prince Alexander of the Netherlands (1951-1884) and his half-sister, the future Queen Wilhelmina, since both were born to a father (King Willem III) who technically was the reigning grand duke of Luxembourg.
The Netherlands: after the the aforementioned Alexander and Wilhelmina, Queen Juliana would count, since she was born in 1909 to a sovereign mother who had reigned since 1890 (enthroned in 1898).
Belgium: the last time this happened was the birth in 1934 of the future King Albert II. Any hypothetical child born to King Baudouin and King Fabiola would have counted, since he (thanks to an abdication) had succeeded to the throne as a bachelor. As it was, the couple was not blessed. Before, Princess Clementine (youngest child of King Leopold II) would have been an example.
Spain: all the children of King Fernando VII, Queen Isabel II, King Alfonso XII, and King Alfonso XIII would count.
UK: the younger children of Queen Elizabeth II were born to a reigning monarch. Before, it was all the children of Queen Victoria; and before, it was all the children of King George III.
Norway: none since the country acquired full autonomy, with a monarchy of its own, in 1905. Can anybody will me in on the history before?
Denmark: none since the reign of King Christian IX. No surprise, given the longevity of the Danish kings. Can anybody fill me in on the situation before?
Sweden: given the longevity of the Bernadotte kings, His Majesty Carl XVI Gustaf would likely not have succeeded to the throne as a young man of 27, and a bachelor at that. As it was, his three children were all born to a reigning monarch. When was the last time before his reign?
Elsewhere, in non-reigning houses (the focus on royal dynasties), one has that the first three children of King Constantine II of the Hellenes were born to a reigning monarch; the youngest two were born after he was deposed. His aunt Katherine was born two months after the accession of his namesake grandfather, King Constantine I. The posthumous Princess Alexandra of Greece and Denmark, only child of King Alexander, doesn't really count -- or does she? Before, it would have been all the children born to King George I and his queen consort, born Grand Duchess Olga Constantinovna of Russia.
Speaking of Russia: all the children of the last czar, Nicholas II, would count; before, his sister Olga was the only child born to his father (Czar Alexander III) after his accession to the throne. Before, it was the two youngest children of Czar Alexander II. Before, it was the three youngest children (all sons) of Czar Nicholas I.
Speaking of empires: all the children of Franz Joseph I of Austria would count. Once again, there would not have been a young accession, but for a renunciation of rights of a parent. The younger children of Blessed Karl, the last emperor, were born after he was deposed. Can anybody fill me in on the situation before?
Germany: the three youngest children of Kaiser Wilhelm II were born after the accession in 1888 (the year of three emperors) of their father. Can anybody fill me in on the situation before? Prussia specifically?
Speaking of German kingdoms: the situation in Hanover has already been covered by King George III of Great Britain.
Saxony: can anybody will me in on this?
Württemberg: all the children of King Wilhelm I. Does anybody know of other examples?
Bavaria: King Otto, younger son of King Maximilian II, was born a month after the abdication of his grandfather (King Ludwig I) and accession of his father to the throne. Before, the two youngest children of the said grandfather were born after his accession in 1825.
Italy: all the children of King Vittorio Emanuele III were born after their father's accession to the throne. Can anybody fill me in on the situation before? Sardinia in particular?
Two Sicilies: all the children of King Ferdinando I would count. Only the youngest child of his successor, King Francesco I counts. But all the children of his own successor, King Ferdinando II, count.
Portugal: all the children of Queen Maria II count. Before, the older children of her father (King Pedro IV, who became Emperor Pedro I of Brazil) were born prior to their father's accession; the younger ones were born after. After, the children of King Luis I were born to an enthroned father. The elder son of King Carlos I was born before his father's accession; the younger (who would become the last king) was born after.
Are all these facts true? Can anybody fill me in on missing pieces? What about France, both royal and imperial?