I didn't watch the video, which may make clearer exactly what the restrictions are.

Also "limited-access lane" is a good term, too, except in AE that subsumes also the so-called HOV (high-occupancy vehicles) lanes, for cars with passengers.

If you need to be sure that buses are included, not only emergency vehicles, then "bus and emergency vehicle lane" is another common phrase:
https://www.google.com/?gws_rd=ssl#q="bus and emergency vehi...