It's even more explicit in the English lyrics: the very first few lines are about buying the balloons and setting them free to fly. None of that is explicitly mentioned in the original German, but the balloons are consistently talked about as concretely existing objects -- if they weren't, what was there to be taken for UFOs? -- so AFAICT the literal interpretation has always been the way pretty much everyone has interpreted it. The ballons are WHY the missiles are flying overhead, but in the beginning, there were actual balloons.

Side note: They come in "a bag", so there must be a phase of blowing them up in between, but that's not mentioned. And since they're flying, they can't be literal "Luftballons" (Luft = air), but must be helium-filled. That's a bigger operation, though, which would takes away some of the song's intended spirit of casual spur-of-the-moment thing -- "We were just fooling around in a toy shop and letting fly a bunch of balloons at dawn" (weird time for a toy shop to be open, BTW) -- so no wonder it's never mentioned.

Also, BTW: If the balloons were taken for "UFOs aus dem All" -- UFOs from space -- then why the fuck would the nations of Earth start shooting at each other over it? Come to think of it, I may already have questioned this way back forty years ago when the song first became popular; I was an SF fan and a stickler for accuracy even then.

But hey, it's a pop song. The lyrics may not make perfect sense, but in that respect they're at least not much worse than most other pop song lyrics.