Learn English – Why does “right” mean “clockwise”

You're right in that you wouldn't find slang in official titles. However, according to Wiktionary, that definition is not slang, but is instead an archaic definition:

(archaic, sometimes used in titles) To a great extent or degree.
Members of the Queen's Privy Council are styled The Right Honourable for life.

In other words, it's a synonym for very. Those who are Right Honourable are honourable to a great degree, more than just The Honourable, but not quite so much as The Most Honourable.

Etymonline says that the word clockwise arose in 1870, much later than clocks. Before that, the word sunwise was used, but it appears to have been fairly rare.

There were words (e.g. deiseil and tuathail) for these concepts in the Celtic languages, since in Celtic cultures the directions clockwise and counterclockwise are quite important, but English seems not to have felt a great need for these concepts, and used sunwise on the rare occasions the concepts came up. All the pre-1850 Google books hits I can find for sunwise or sun-wise are describing Celtic, Hindu, ancient Roman, or American Indian customs. The earliest reference I can find on Google books for sunwise is 1775.

From The origin and history of Irish names of places, by Patrick Weston Joyce (1875):

Deisiol [desshul] is another derivative from deis, and signifies towards the right hand, or southwards. The Celtic people were—and are still—accustomed to turn sunwise, i.e. from left to right, in the performance of various rites, some of them religious, some merely superstitious, and the word deisiol was used to designate this way of turning.

If sunwise had been a common term at the time, or if there had been another commonly-understood word meaning clockwise, the author would not have felt the need to explain it by "i.e. from left to right".