Note that this information is drawn from a variety of EU works and may include spoilers for certain stories
Interestingly, royalty is connected to both sides of the Solo-Skywalker union.
Han Solo is possibly descended from Berethron e Solo, a king of Corellia who introduced democracy to the system. Han, through C-3P0 used this to claim that he should marry Leia Organa before a member of the Hapan Royal House could make a claim. At the end of The Courtship of Princess Leia, C-3P0 finds that the link between Han and Berethron may be a pretender to the throne, so Han's 'royal status' is questionable. However, if Han indeed was the rightful King of Corellia, then his children surely would have been royalty.
Leia Organa Solo, as you noted, was the daughter of Padme Amidala, once a Queen of Naboo. As she was democratically elected to be Queen, it is unlikely that her children would automatically be royalty. Note that Padme's parents, Ruwee and Jobal Naberrie, do not appear to be especially prominent in Star Wars Episode II. However, after Padme's death, Leia was adopted by Bail Organa and his wife, Queen Breha Organa. As Bail Organa did not continue as head of state until his death, later becoming a Senator, it is unknown if his grandchildren would have any royal status. As Alderaan was destroyed by the Empire, however, any royal line there effectively ended.
The Solos had three children: Jacen, Jaina, and Anakin. Of the three, Anakin, who died as a teenager, is the only one without explicit royal connections on his own. Jaina Solo eventually married Jagged Fel, who went on to become emperor of the revived empire, founding member of the Fel dynasty. As such, presumably Jaina had some form of royal title as his wife (Fate of the Jedi series, various comics).
Finally, Jacen Solo was the lover of Tenel Ka Djo, the Queen of Hapes and ultimately fathered a child, Allana Solo, who was adopted by Han & Leia. Had Jacen openely married Tenel Ka, he would have had a royal title, and had his parentage of Allana, the Chume'da of the Hapan system, become known, he would have been an important figure in Hapan society, despite it having a matriarchal society (Legacy of the Force series).
In summary, it is not clear if Leia's clear royal heritage was passed onto her children, but they did have some royal connection regardless, whether through their own relationships, or through a possible parental royal link.