The client would download the first 1KB for an icon, 4K for a thumbnail and the whole image for the full size. The best part is that all of those are actually the same file, not separate embedded thumbnails inside the file.

The client could decide BY ITSELF to download only 2K for a thumbnail because it's currently on 3g.

Where are you getting 1KB and 4KB from? The compression is highly dependent on the image so there's no way to know the quality of the thumbnail image.

Maybe a future version of the format would specify useful cut-off points for browsers to make the decision inside the header. Then the browser can decide to download 4K version on wifi and 1K version on 3g.

You'd need to set the thresholds in the file manually, based on the artists judgement of quality.

It would take a few extra seconds when creating it, but could save both bandwidth and create a better user experience on mobile.

This will have to be automated or it's going to be really annoying for development. To eliminate that, maybe the file format could incorporate some internal tags for good size thresholds, or even just a handful of coefficients at the beginning for a simple curve approximating the byte-to-resolution mapping.