A friend told me about this bother yesterday - they use the web interface to read their email whereas I normallly use Thunderbird as my email client.
Anyway, I looked into it last night and I found out that VM on Monday, without warning, decided to force people to use the new VM-O2 ID system at the VM login screen.
So if you logged out and cleared your browser's cache, you'll come across the new clusterf==k of the login prompt whereby you have to create this new VM-OS2 ID account from your existing VM one. You have to verify you own the VM ID by clicking on a link that they send you to an alternative email address (in my case Gmail). God knows what happens if folk dont have a secondary email address. Obviously they assume most are Android or iOS phone users.
The second thing I realised is that when they send you that link, make sure you open it on the exact same browser and device that you are registering the 'new' VM-OS2 ID account. For some reason the browser cookies involved are not being passed to other browsers and devices - hence the reason you get stuck in that loop. Don't mix phone and computer - choose one and stick with it until the VM-OS2 ID has been created Once it is and you log in, you will be at the 'My Virginmedia' webpage and you can then access email from the 'Quick Links' near the bottom of the page.
Good luck.