Here is the public view https://drive.google.com/file/d/1s48yut19UH-IShu_ejUnnGmZVwQQLbmv/view?usp=sharing
Here is the editor view where I changed the route number to 677.https://drive.google.com/file/d/103UoFGiQ5G8UvuBTvqiwpFrImF9tfYHC/view?usp=sharing
Note I saved my changes but how do those changes take effect?
Hi,
The changes take effect immediately, but I imagine you are looking on the OpenCycleMap layer. This is only updated something like every week to 10 days. You can see that it shows as 677 not 667 on Andy’s (SomeoneElse) map (https://map.atownsend.org.uk/maps/map/map.html#zoom=15&lat=53.82988&lon=-1.49241), albeit less obviously so, as this map is more designed for walkers than cyclists.
Cheers,
Jerry
So what you are saying is I need to check back in 10 days time to see if the update has happened?
Thanks.
The update happened to the main database, as SK53 already checked. you do not need to come back to verify that. When you have pressed save and the editor indicates no errors, you can be sure your update is made.
But there are many data consumers out there, from the “default” carto-osm rendering over the cycle map to many styles for Garmin, OsmAnd, etc. Each has its own update cycle. This can be from a few minutes to once a month or even less frequent. You do not have to check each of them individually
What exactly does that mean? Is another user updating with the incorrect data?
677 is the correct route number. I have been marking it on the ground as such. 
OS maps show the correct route number also: https://osmaps.ordnancesurvey.co.uk/53.80205,-1.49010,14 (To see this apply the National Cycle Network layer).
No, as other people have said, you’ve successfully updated it. You can see the whole route in OSM at https://www.openstreetmap.org/relation/9232228 . What people are trying to explain is that not every website that uses OSM data updates their maps at the same time. You can see that at https://www.openstreetmap.org/relation/9232228#map=19/53.81449/-1.48103&layers=C . The OSM data (on the left) says 677, but the Cycle Map (on the right), which hasn’t updated yet, still shows 667. The cycle map will update at some point, but you and I can’t control when it will; only the person who controls the server that hosts that map style can (that map style is not hosted on OpenStreetMap servers).
Ok thanks for the explanation :). I am fairly new here so forgive the ignorance of how things work.
I have emailed Andy Allan (https://www.thunderforest.com/) as there is a link to him bottom right corner of the map.
I assume he controls the map data for the cycling layer.
SOLVED
Looks like the change has gone through now. Thanks for the help. 