In theory you just need to add the 'addr:postcode' tag to the relevant house or street. That said, OSM's geocoder has a few issues with postcodes at present: h

In theory you just need to add the 'addr:postcode' tag to the relevant house or street. That said, OSM's geocoder has a few issues with postcodes at present: https://trac.openstreetmap.org/ticket/2497 https://trac.openstreetmap.org/ticket/2497 . The second para of lonvia's final comment is the crux - "[postcode updates are] a very expensive operation and so the centroids don't get updated and postcodes go stale with time".