Here are some graphs derived from our service that compares road names in OpenStreetMap with those in the Ordnance Survey's OS Locator product.
This first graph shows how many road names are missing from OpenStreetMap that are in OS Locator with the 408 districts/boroughs organised with the ones with the least missing on the left and the most missing on the right. Do be aware that the size of authorities varies very considerably from the Isles of Scilly with 57 roads to Leeds with 12,000! Clearly there are a small number of larger places with where a lot of work is required.
This next graph shows the percentage completeness of these same authorities. The colour bars match with the thematic map. Nearly half the authorities now have 75% completeness and do be aware that many of the missing road names are likely to be for very minor rural roads/tracks or for small cul-de-sacs in town. As such an authorities with 75% completeness is likely to be very usable for most purposes.
This final graph shows all the 253,000 road names in OS Locator that are currently missing from OpenStreetMap as a cumulative version of the first graph. The most complete authorities are again on the left. It shows that one could complete the 'best' 204 authorities by adding 50,000 roads but to complete the others would need a further 200,000 roads with is a lot of work to do manually.
Personally, I am starting to think about a bot that could add a lot of these roads automatically for the less well covered areas using OS VectorMap District an OS Locator. The bot would do what it was sure about for an area as instructed by a mapper who would then do a cleanup job of the area afterwards.