My apologies. I realise it seems obvious, but when I checked this on my dev site, I actually didn’t have that box checked. I thought it might be fairly easy to miss.
Thank you for the links to the website in question. It looks like the issue is that the reCAPTCHA JavaScript is not loading on that page.
Having a look at the code, it appears that I attempt to get the ID of the page which is set to be the BuddyPress ‘registration’ page, and only load the JavaScript there.
Is it possible that this setting is not correctly set?
I’ll keep checking to see if there might be something else which is causing this.