Looks like it auto selects based on network speed. When I load the page it starts playing 960x540 and after a few seconds changes to 1280 x 720
Here found the info page.
https://www.bbc.co.uk/iplayer/help/questions/programme-availability/hd-live
If your location is set to any of the English Regions you won't be able to get BBC One in HD, so you'll need to change your location to England, Scotland, Northern Ireland or Wales if you want to watch BBC One in HD.