The problem was resolved by enabling virtualisation in the BIOS, but it was not immediately clear to me that there were two settings that needed to be enabled so I thought I'd share.
Here's a shot of the relevant BIOS screen (it's one of the sub menus under System Configuration):
"Virtualization Technology" was already selected in the BIOS, so I couldn't understand why HAXM wasn't installing.
Then I realised that Data Execution Prevention needed to be active (as indicated by the red arrow in the screenshot above). Data Execution Prevention is the name given to Intel XD (I advise grabbing a coffee if you're inclined to follow the link) and to be fair, the BIOS feature is mentioned by name in the tail end of these instructions, which i guess just further proves the old maxim RTFM.