Is there any way to get contour lines wth OpenStreetMap's Overpass API? I can see them in the cycle map - but I can't seem to get this data at all through Overpass.
There is NO way to get contour lines with the Overpass API.
Overpass API only has access to the OpenStreetMap Server, and that contains no dense elevation data to create contour lines.
The contour lines in the cycle map of openstreetmap.org are generated from elevation data of other sources, like SRTM, Aster or else. The author does not reveal much about the styling of his map.
1At least not on the official instance on overpass-api.de. When setting up your own instance you could load data which reflects contour lines in some way. Yet, there's nothing available to support this approach out of the box.– mmdJul 5, 2017 at 5:02
As already answered, there is no contour lines data in OSM, but only some elevation data for mountain peaks and other points.
For your information, you can find how are generated the contour lines of OpenTopoMap, another OSM-derived map, in this page.