I think strict reading of standards would suggest the answer to your question is "no."
PADI has fairly specific requirements on crediting the Nitrox course toward an "Adventure Diver" or AOW certification, including studying and retaining all materials required for the PADI Enriched Air Diver course. (Adventures in Diving Program Instructor Guide, p. 1-5 under "Exceptions"). Specifically, student divers must study the PADI nitrox manual, complete knowledge reviews, and watch the PADI nitrox video (Instructor Guide, p. 2.2). Furthermore, the student must attend an instructor led nitrox lecture, and in order to conduct that lecture, the AOW instructor must also be a PADI Nitrox instructor. Moreove, the instructor must directly supervise the student on all certification dives, so if the instructor isn't present during the TDI course, you'd be out of luck.
Some shops will offer a dual certification. My LDS offers both PADI and TDI classroom courses together, but charges a little extra for the actual TDI cert because it requires dives.
But if an MSDT wants to chime in with the "right answer", please do. I'm just a lowly DMC with a manual.