Cycling is currently banned in the High Street, resulting in some cycle parking being difficult to access legally and some useful routes across town (such as Union Lane to Baker Lane) being cut off.
The ban was not enforced between roughly 2015 and 2018 because some required signs were not put back after restoration of the market places. We do not know of any significant increase in injury incidents during that time. Of course, the ban is widely flouted despite being enforceable again because it looks like a road, with a marked carriageway, and there is cycle parking provided on High Street.
Access to the south end between SMP and Purfleet to avoid the narrow Queen Street (or worse, the A148) is probably more vital than the northern section where King Street isn't a bad alternative.
Government policy since 1987 has been that such pedestrian areas should be pedestrian-and-cycle areas unless there's evidence of problems and provision of alternative routes. Norwich's pedestrian areas are now mostly pedestrian-and-cycle.