Stan has corrected me. I mis-read the Reynolds timeline, but there they state that the first 531 tubing was introduced in 1935. They were making butted steel tubing for bicycles, as early as 1902 (or maybe as early as 1898, as that's the date the "Patent Tubing Company Ltd." was formed according to the timeline, but that's not clearly tubing for bicycle frames).
I'd forgotten about Accles and Pollock tubing...how could anyone forget a name like that?

Anyhow, to go back to the OP's question: it would seem that the first use of chrome-moly tubing would have to something made from A&P's Kromo tubing, which was in use for bicycle frames as early as the '30s, including the CCMs that T-Mar mentioned, but an earlier example may come from a UK builder such as Hobbs of Barbican, Thanet or Rattray's "Flying Scot".
At least that's where I'd be looking for the final word in "who's first?".