I'm not defending the the removal of the headphone by any means, but you are unlikely to experience any lag with the lightning cable. It's just a pcm bytestream getting out of the phone into a dac in the headphones. The latency won't be any different than what already is.

If you are talking about their new wireless headphones, It will probably have some delay but probably not even close to bluetooth. They could just modulate that same bytestream to some ghz radio frequency without adding anything remotely as over engineered and cumbersome as a bluetooth stack. I'm pretty sure those airpods are not bluetooth compatible.