Still lacking info about everything in the chain, but a couple quick thoughts..

Broadcast standards call for a max lipsync delay of 2 frames (66ms). More than that will be noticed.

Assuming the audio goes through the mixer then to the black magic web encoder, there shouldn't be any audio delay.

Delay to the video can be introduced in several places:
- In the camera. Some (cheaper) cameras add 6 frames of delay, or more. Better cameras add a frame or less
- In the video switcher. Cheaper switchers add several frames. Better switchers add none.
- Scalers/converters. These typically add at least a frame or two.
- Frame sync. These add a frame of delay, at a minimum.
- Encoders. These can add lots of delay.

Generally speaking, a DA (or splitter) will not add any delay.

A web encoder, like the black magic encoder, will add quite a bit of delay, however, it is designed assuming that the video and audio coming into the device are in sync, and it should keep them in sync, with equal delay, throughout the process.