Britain began to become involved in the trans Atlantic slave trade in the early 1600s and slavey wasn't abolished in the British Empire until 1833, giving us a fairly large window. Did any member of the British royal family own slaves from Africa or elsewhere?
I don't know if the Royals ever directly owned slaves - in the sense that they had slaves working directly for them.
However, the Royals were major shareholders in the British Corporations that were established to conduct the slave trade - buying slaves in Africa and selling them in the Americas. Documents from the early 1700's show that literally "all the Royal family" was heavily invested in the South Sea Company (which crashed in a notorious market bubble) - including Kings Charles II, James II, Queen Anne, and George I. The South Sea Company's primary purpose was to deliver slaves to the Spanish Empire.
Also note that Thomas Jefferson refers to the Georgian Royals' investments in the slave trade in his first draft of the Declaration of Independence, where he condemns George III for waging "cruel war against human nature itself, violating it's most sacred rights of life and liberty in the persons of a distant people who never offended him, captivating and carrying them into slavery in another hemisphere". George III and his son William were against the parties in Parliament in the late-1700's who favored outlawing the slave trade - no doubt because they were to lose income if it happened.
Source - "The Slave Trade" by Hugh Thomas (1997)
As a follow-up question, if slaves were linked to the royal family: were they personal property of individual member(s), anonymized through business dealings (similar to shell corporations, maybe gifts or sent to faraway estates), or crown/state property? And is there a legal basis for the status they did or didn't have, or just custom?