How did antonyms "ungodly" and "unearthly" come to have the same meaning and usage today?

If you look a little further in your Etymonline reference, you see that unearthly acquired its pejorative usage around 1802:

Sense of "ghostly, weird" first recorded 1802.

Partridge records the first reference of an unearthly time from 1865.

I can find no citation to explain these particular transformations, but you should remember that unearthly also easily describes things that are ungodly. Beings that were not considered to be part of this world were in the domain of the "spirit world" — which is believed to exist by some even today, and which includes the malicious as well as the sublime.

I'm not sure you've read the debate properly. Neither word means 'outrageous'; the basic meaning in each case is exactly what you would expect from the etymology: 'evil or dishonest' and 'unlike anything on earth' respectively. There being less demand nowadays for these precise meanings than there was (except in parts of Nevada), the meanings have been extended. Music that is an ungodly racket is deeply unpleasant; it might also be described as an unearthly racket exactly as it might be said to sound like nothing on earth. These are certainly used (and in the dictionary) as synonyms for 'unpleasant' (originally jocularly). But if you say that, because of this use, the two words mean the same you will rapidly reach the conclusion that all adjectives mean either good or bad.