It's late evening where I am, so I'll thank you all again and look at this in the morning.

It's late where I am now, but I'll look again tomorrow at the suggestions that have been made. Many thanks to you all.

The missing box should be between the password box and the sign in button. The device that I am using is a Lenovo Yoga 3 pro 1370. As I have said, the missing box has displayed for a while in the past, but not any more.

I'll provide comparison shots for reference.

Why is the Recaptcha checkbox missing in shot #2? The gstatic.com domain is blocked, that is the only difference - even the browsing session is the same and the experiment and result can be reliably repeated.

I used NoScript for blocking, but this could also be done by antivirus (or an advanced firewall or child protection/website censoring tool) or perhaps some Firefox setting that prevents content from loading from a specific (third-party?) website. I tested Private Mode and tracking protection does not prevent the checkbox, at least. In the settings I can only find blocking for third-party cookies, which should not affect the checkbox.

Anyone else have insight into which Firefox setting(s) could cause content rejection or blocking?

I assume your desktop is on a cable connection and your laptop is on a wireless connection. Have you tried connecting the laptop via cable (in case something is filtering your wireless traffic), preferrably the same cable/socket your desktop uses?

PS. If you are feeling crafty, you can go to inspect the element(s) of a page in Firefox by right-clicking on the page and selecting Inspect Element (see attached image #1).

Now, with the gstatic.com domain blocked, the website will not fully load and if you search for "recaptcha" (see image #2), you only get 3 hits - the groundwork is there but nothing comes through, the data is simply not loaded.

Then, if we unblock gstatic.com and repeat the experiment, voilá: 20 hits and the box is shown (see image #3). The data has loaded.

This would confirm whether or not the data for the checkbox exists in the first place, i.e. is gstatic.com blocked or has it simply not yet loaded or been rendered.

Just to explain the reason for my confusion.

I see a login box with no reCaptcha, but also with no yellow warning about having to prove I am not a robot.

However I am not testing from Windows at the moment, and will likely be in a different country to you.

That's an intriguing finding; it's true that verification may only be requested of particular clients, e.g. a "foreign" IP (even a Google search may require captcha when using Tor-masked connections).

Phoxuponyou said

PS. If you are feeling crafty, you can go to inspect the element(s) of a page in Firefox by right-clicking on the page and selecting Inspect Element (see attached image #1). Now, with the gstatic.com domain blocked, the website will not fully load and if you search for "recaptcha" (see image #2), you only get 3 hits - the groundwork is there but nothing comes through, the data is simply not loaded. Then, if we unblock gstatic.com and repeat the experiment, voilá: 20 hits and the box is shown (see image #3). The data has loaded. This would confirm whether or not the data for the checkbox exists in the first place, i.e. is gstatic.com blocked or has it simply not yet loaded or been rendered.

When I attempt to log in and allow, via 'no script' gstatic.com and reload the page the page reloads without the yellow band, Then when I attempt to log in the yellow band is back, and then the only thing forbidden on 'no script' is 'Allow all from [null] When I allow that and 'clear recent history' and try to log in the yellow band is missing and I can then log into the 'My Virgin Media' page and from there I can select the email option at the top and can log into VM's web-mail with any problem.

Well, that's how it is at the moment but whether it will always be possible I am not sure. In any case this does not solve the original problem.

I have disabled 'No Script' and when I right click on the 'problem' page and select 'Inspect element (Q) it opens a whole bunch of stuff that I know absolutely nothing about. Could there be anything connected to this that could be causing the problem ?

I have now enabled 'no script' as I was seeing a lot of advertising pop ups, but the problem persists.