Now you’ve put a kink in Water Street too.

If the LCN tagging is not continuous, split ways and add more lcn=yes tags. There’s no need to change geometry or change cycleway links to do that.

Those cycleway links were in place because the span of a cycleway as it crosses into a road carriageway is clearly different from the cycleway as it continues outside the carriageway. Hence the separate tagging and separate way.