There is still a discrepancy between OSM and cycle.travel. In Skelmersdale on Southway, cycle.travel suggests a route that is going the wrong way down a busy one way dual carriageway. OSM shows the traffic flow correctly. This has been the case for some months now. OSM did have it wrong but has been corrected.
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The OSM data was changed last month (August) just after cycle.travel took its last map update, so c.t hasn’t reflected it yet. It will show through in c.t in a week or two when the next map update happens.
cycle.travel must have updated now because my trial local route finding suggests what I think is the best route. Great work that cycle.travel is so up to date!