Currently, if you try to save a battle replay on Firefox iOS, then clicking the Download button results in nothing happening.
You can long press on the download button to open the raw HTML in a new page, but there's no way to save this content (except for copy and pasting the entire HTML document into somewhere).
The only workaround I can find is to open the same link in Safari, then press the download button (which works correctly). Not sure if this works in Chrome too.