The Google elevation API provides a vertical accuracy of 30 m and there is no other open elevation service providing more accurate data available. During the development of Predictive Tracking we noticed, that this cartographic data is not precise enough and we developed a self-hosted elevation service (vertical accuracy of 1 m), you can use for your own projects under MIT license. You either host this service yourself or just use our API.
Mapzen terrain data: https://registry.opendata.aws/terrain-tiles/
GitHub documentation and source code: https://github.com/racemap/elevation-service#api-usage
hosted elevation service: http://elevation.racemap.com/v2/?lat=51.3&lng=12.5
You can download portions of the data. The data is organized by latitude/longitude where each chunk is 1x1 degree. Eg. for downloading the UK data you need to know the lat/lng extent of the UK and download all the necessary files. Example:
aws s3 cp --no-sign-request s3://elevation-tiles-prod/skadi/N51/N51E000.hgt.gz /path/to/data/folder/N51
Make sure to reproduce the file structure when downloading, e.g.
./N51/N51E001.hgt.gz. The whole dataset is just ~200GB, so maybe it is even feasible to download the whole thing.