This question comes nearly once every week.....
The ATEM itself (TVS and 1ME models) has a one frame maximum video delay. This is not noticeable. But often, you have an additional delay from the cameras, depending on quality and price. Professional studio cams have a delay of nearly zero, customer cams one, two, maybe more frames, because if internal image processing. This is normaly not a problem, because their outputs are primary made for playback a recorded video and the audio is also delayed inside - but not in live environment. When you combine the delayed video with non delayed audio coming from a audio mixing console, it will be noticeable.
Some people say, feed audio also via a camera to add also an audio delay. I would prefer a delay between mixing console and ATEM using an audio effects unit (e.g. from Behringer).
You can research the right value for the delay by recording clapper board and compare audio and video inside an editing program. In nature, Audio is allways behind video, so you don't need to set a total exact value, but rounded to the beginning of the next frame. (Values depending on framerate)
Ater this, you schould have synchronous audio and video at all ATEM outputs.
Part two:what happens behind the ATEM ?
I don‘t think, that the hypedeck adds additional dalay - But if, its only on addition part in the complete „delay chain“.
Your monitors also have a video delay, and projectors more...
So when the viewer is able to see the real live scene and the monitor/tv the same time, there will be allways a noticeable delay. When the audience hear live audio and took to monitors, there will also be a delay - no solution ! Look at big and expensive tv shows. The video wall ist allways delayed to the real scene.
Theoretical, you could send the delayed audio to the audience, so monitors and audio are in sync, but you drive all actors and musicans crazy, because of huge audio echos....(They hear themselfes with delay)
You can produce a lip-sync recording and broadcasting. You can‘t really bring a lip-snc singnal to the audience.
At last, monitoring in the control room. Using the audio section of the control tv-monitor schould work fine. There are built for synchronous output of video and audio
When using the audio from the audio mixing console, please use the signal behind the delay unit for the control room but you may have a small delay (from the control monitor). Live with it or use a second delay unit for compensating this...
I hope, this helps for your understanding. A last word to „genlock“. There are cameras, which can synchronized to the timing of the ATEM and the other cameras. These cameras have faster processing, because there are made for live usge, so their internal delay is small and the delay inside the ATEM is nearly zero. But these cameras are other price classes.....