This sounds like an error in the OpenStreetMap data. If you click the "edit" button in the bottom right corner of the cycle.travel map, you'll be redirected to the OpenStreetMap editor where you should be able to mark the road as private (From looking at Streetview, it seems like there are some houses at the start of the road that should still be accessible, so you'll want to split the road into two at the location of the gate, and only mark the after-the-gate section as private)

The great thing about this is it will benefit all apps and websites which use OSM data, so you'll be improving the map for everyone!