Is "Under Weigh" Correct? Or is "Under Way" the Right Weigh?

Yes. Spelling matters. (More about that at "Soh Whut ef i cant spel?"
(Only a Looser Chicks Four Spilling Mystics).)

Is it "under weigh," or "under way?"

• British English:
  "UNDER WEIGH"
  World Wide Words
  • "Under way" is now standard usage
  • There's more about weigh and way, starting with Dutch in the eighteenth century
• American English: "under way, underway, under weigh"
  The Columbia Guide to Standard American English, 1993
  • "...Under way and underway are both now Standard spellings of both adverbial and adjectival uses."
  • There's more about the phrase, of course

So: on both sides of the Atlantic, it's "under way."

Why bother with a post about this? I looked up current standard usage for another blog, and decided that I'd share what I found.