There are many such words in a fiction literature, including Jabberwocky, Cthulhu, Kilrathi, and many others. Of course, the authors intended to make these words appear most uncommon to an English reader, and it must have been a deliberate attempt to make the pronunciation unclear.
Being an English learner, I, most likely, have read those books translated to my language, so it is very possible that my pronunciation is somewhat spoiled (even if my language uses Latin alphabet).
An obvious way is googling for each individual word to see how it is suggested to pronounce each one. However, this is not the case if you are involved in a spoken discussion. You simply have no possibility to search.
So, is there any simple, mnemonic rule how to pronounce an arbitrary artificially created word?
Update: I see close-votes on this question, so I have to explain myself.
Not knowing how to pronounce a word may lead you to an embarrassing situation.
Actually, a comic below illustrates very well what happens if you pronounce R'lyeh wrongly:
image from here