Hi - the long-standing convention in OSM, dating back to 2011, is that the National Byway is network=rcn. This is due to its Regional Route-like character; the fact it's not truly a national network but just covers some regions of the UK; and for differentiation from the (Sustrans/WWCT) National Cycle Network. I've retagged the superroute relation accordingly.

(writing as someone who has a spare National Byway sign above their desk ;) )