Very much related to this question Can a Kingsguard member be sent to the Night's Watch? but now the other way around, do we know of any Night's Watch brother that became Kingsguard?
Almost certainly not and I can find no cases of it ever happening
There are no recorded cases that I can find of a Night's Watch brother being released from their oaths and joining the Kingsguard. In fact I can't really think of a case of a sworn brother being released from their oath. We have Jon Snow but he pulled the whole loophole trick in the show and we're yet to see what happens in the books so we can discount that. However, Stannis does offer to legitimise him and free him from the oath, though it is not clear if he actually can do that or is doing it because he has the army and power to enforce it.
"As you wish. But consider quickly. I am not a patient man, as your black brothers are about to discover." Stannis put a thin, fleshless hand on Jon's shoulder. "Say nothing of what we've discussed here today. To anyone. But when you return, you need only bend your knee, lay your sword at my feet, and pledge yourself to my service, and you shall rise again as Jon Stark, the Lord of Winterfell."
A Storm of Swords, Jon XI
The only known case of someone else being released from the oath is Samwell Tarly at the end of Season 8 but it happens off screen. We can assume the new King released him so it would appear that kings can do this but usually don't as the men of the Watch are usually less than stellar. Sam is also made Archmaester so this might have something to do with it as well.
The oath explicitly states "I shall live and die at my post." and so would make it clear that one cannot simply leave, as we see deserters are executed.
Night gathers, and now my watch begins. It shall not end until my death. I shall take no wife, hold no lands, father no children. I shall wear no crowns and win no glory. I shall live and die at my post. I am the sword in the darkness. I am the watcher on the walls. I am the shield that guards the realms of men. I pledge my life and honor to the Night's Watch, for this night and all the nights to come.
In fact the Kingsguard oath was modelled on the Night's Watch one and so it would appear whilst they have common elements they probably conflict as Kingsguard members serve for life.
It was Visenya, not Aegon, who decided the nature of the Kingsguard. Seven champions for the Lord of the Seven Kingdoms, who would all be knights. She modeled their vows upon those of the Night's Watch, so that they would forfeit all things save their duty to the king.
The World of Ice and Fire, The Targaryen Kings: Aegon I
Lastly, it is worth noting of those Kingsguard that were sent to the Night's Watch most had lost their king and broke the oath by betraying said king. And the odd one out broke the oath by fathering children. Breaking the Night's Watch oath results in death not being released from it to join a more prestigious organisation.