For example, EuroVelo 12 north of Tain is shown in OSM as a red line but the relation colour is green. It would be helpful if EuroVelo routes can be displayed in a different colour when they are not part of the UK National Cycle Network.
I guess the first question is “which map on which website”?
A challenge with cycle routes is that they often overlap, and if you have fixed solid colours with overlap you’ll get a confusing map 
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It’s the cycle map layer on OSM. NCN routes are shown in red and EuroVelo routes display as red even though the relation colour is green. I’m looking for a solution where EuroVelo routes are not part of the NCN.
The website you want is Waymarked Trails. This focuses specifically providing information about different routes.
It is perhaps not as pretty as the cycle map layer, but combining functionality and prettiness is rarely straightforward!
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Or, of course, several others - here’s cycle.travel, for example:
The NCN sections have both an NCN number patch (red) and a EuroVelo one (dark blue), while the others just have the EuroVelo one.
The “Cycle Map” layer shown on openstreetmap.org is in fact OpenCycleMap, which is a rendering maintained independently by Andy Allan. Andy’s website is thunderforest.com which has a link for contact details at the bottom of the page.
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