In Chrome on Android (I tested with v52), you get a prompt saying "Chrome needs storage access to download files". You can choose "Update permissions". This prompts "Allow Chrome to access photos, media and files on your device?" to which you can tap "Allow". Then your photo is downloaded to the 'Download' album.
But on iOS, both Chrome and Safari simply load the image up in another tab. (And as mentioned, Samsung Internet on Android blocks it).
I tested using: https://github.com/poshaughnessy/image-download-tests
If it could work in the same way as Chrome on Android, consistently across other browsers, I think this would be pretty good, at least as a short term solution.
Although I do agree with @simevidas that this method is a bit of a hack. (Imagine a native developer coming to the web and asking how to save a photo, and you answer "well, you just generate an anchor tag element with a 'download' attribute and set the src attribute to your data-uri and then simulate a click on it!").
My own instinct would be to prefer adding a simple image option to an existing File Saving API, rather than having its own separate one, but I guess this will take more thought...