tolkiens legendarium - How did the Black Speech come about?

It doesn't seem to be related to any other documented languages. Being endemic to Sauron's realm, with him persisting through the Ages, it would not likely have diverged naturally from another language anyway.

There is this related question, which deals with how Tolkien the author arrived at the Black Speech. The answer includes this quote:

The Black Speech was not intentionally modeled on any style, but was meant to be self consistent, very different from Elvish, yet organized and expressive, as would be expected of a device of Sauron before his complete corruption.

(emphasis added)

That answer doesn't address how the language came to be; but the emphasized portion implies Sauron developed it himself. That's certainly reasonable though I'd be curious when and why. Did Tolkien expound on that anywhere else?

Answer

Appendix F of The Lord Of The Rings ("The Languages and Peoples of the Third Age") has this to say:

It is said that the Black Speech was devised by Sauron in the Dark Years, and that he had desired to make it the language of all those that served him, but he failed in that purpose. From the Black Speech, however, were derived many of the words that were in the Third Age wide-spread among the Orcs, such as ghâsh 'fire', but after the first overthrow of Sauron this language in its ancient form was forgotten by all but the Nazgûl. When Sauron arose again, it became once more the language of Barad-dûr and of the captains of Mordor. The inscription on the Ring was in the ancient Black Speech, while the curse of the Mordor-orc in II, 53. was in the more debased form used by the soldiers of the Dark Tower, of whom Grishnâkh was the captain. Sharku in that tongue means old man.

That's the answer, then; Sauron developed it towards the end of the Second Age, when he was achieving domination over much of Middle-earth. He didn't succeed in enforcing its use among all his creatures, but (especially toward the end of the Third Age) his more powerful minions spoke it as their primary if not their exclusive language.