You would need a 1-1/4 p-trap to fit that trap adapter, even if you find the nut. So your thoughts were correct, I think.
How big is the thread that trap adapter is screwed onto? Some could probably tell by the photos, but I can't. You will need a trap adapter that fits that thread, and you will need a p-trap to match the trap adapter.
I would unscrew that trap adapter piece and see how big the thread it is screwed onto is. You will be buying a new trap adapter anyway.