I am unable to duplicate your problem probably because I do not have HDMI on my desktop. I have however used NVDA on laptops and connected them to HDMI projectors without problems. My suggestion came from my experiences with USB sound cards. Setting a specific sound card helped mitigate the problem that NVDA would switch to a USB sound card when that card was inserted. HDMI was doing something similar hence I thought the same solution would work.
It appears the solution failed, sorry about that.
I did try running a presentation in slide show mode and NVDA began to speak without any effort. Pranav