The council is proposing to re-design the junction to allow buses and taxis a convenient short cut (shown in red):
Why won't cyclists be legally allowed to use the new facility? Taxis will.
I suspect that the council is considering converting the pavement on the north-west side of White Street into a pedestrian-cyclist shared use facility. While that will work for slower cyclists, a lot of pedestrians use pavement at this point, so there will be conflict between pedestrians and cyclists hurrying to work.
Why not both a shared cyclist/pedestrian route, for the slower cyclist AND a shared cyclist/bus route, for the faster cyclists? Cyclists in a hurry can easily reach 15-20 mph, they won't hold up the buses over such a short stretch.