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.