I have deleted a couple of bike paths that I had previously added using the iD editor. They do not appear any more when using the query button, but when I check on the Cycle Map layer the blue dashed lines are still visible. Does it take longer for the path to be removed on the Cycle Map layer?
This was covered in a thread that was last updated today!
The cycle map tiles are rendered by a third party. It looks to me as though they have a weekly cycle for re-rendering the tiles.
Please note that, except at the highest zoom levels, even the standard layer tiles an lag significantly.