No, there are no known incidents of royal babyswapping (in Europe, at least). I discuss this in a previous answer to Did royalty ever "baby swap" a newborn daughter to fake having born a male heir? However, the show is also inaccurate - unsurprisingly, given its tagline! It was not the norm to give birth publicly in any court except the French, specifically that of Louis XIV and the few that followed him: Louis created a mode of life for the French royalty that put them constantly on display to bolster their status as the most important people in the country. I've written about this in response to How, in practical terms, did the creation of Versailles "domesticate the nobility"? and Louis XIV created in Versailles an everyday ceremony of clothing of the king in which the highest French nobility took part. Why did he do it? What did the nobility think about it?

It's likely that The Great is deliberately taking points from Sofia Coppola's Marie Antoinette, which depicts this kind of public childbirth. It certainly takes a lot of aesthetic cues from that movie, from costume design to modern music.