Location, see: https://cycle.travel/map/journey/502411
It is a complicated situation, the area is under construction. Two tunnels are being constructed under the railway. This reconstruction will take until 2026, so it will be subject to constant change. That there will be regular inaccuracies on OSM is inevitable.
Two notable issues. First, the map on Windows computer is different from the map on Android phone.
Windows computer
Android phone.
Second, where CT's cycle route deviates from the cycle path, it follows the route as indicated on OSM. Currently, however, this stretch of cycle path is as indicated on CT. Incomprehensible to me is how CT 'knows' the correct situation. Incidentally, in a few weeks, this too will be different.
Both screendumps where made this afternoon.
Comments
Behind the scenes, cycle.travel has three independent maps: the raster display map (which is what you see on the website), the vector display map (which is what you see in the app), and the routing network map (which is what generates the routes on both website and app).
They’re all made from the same revision of OSM source data at pretty much the same time, so they should show the same data.
However, caching can sometimes get in the way and show you old map images rather than fetching the latest, particularly in the case of the raster map. This might be your browser cache or one of the two caches that cycle.travel’s website uses to keep responses snappy. This will clear up with time – if I look now then the journey you linked to now shows the current map, for me at least.