Yesterday I experimented with the Galaxy S8Plus-Fire Stick-Samsung TV combination, and the delay was minimal and negligible, I estimate it at 0.1-0.2 seconds.

There is another problem with a Fire Stick type of implementation: the encoding - IP - decoding delay is significant, enough to keep the mirrored display behind a disconcerting amount. For that reason, you really want a fast serial interface dumping the screen contents to the mirrored screen frame buffer, if the device can't have a true HDMI output.