Hi there. I'm Liam Proven, the software director of Simplicity Computers, and I put the Homekey together with the assistance of my colleagues.
It isn't a new OS. It's a bootable USB stick with a custom installation of Linux Mint 9 LTS on it. (The current version includes a newer, updated Linux kernel, and we are currently working on a Mint 13 LTS release.)
On top of Mint, there is our own custom front-end, the Simplicity Envelope, complete with a comprehensive set of video tutorials and a printed, illustrated, colour handbook. We also operate a customer support hotline which is popular and well-regarded. The Homekey is our latest product and it follows on from a line of PCs - desktops, touchscreens and laptops - which we have been selling since 2009, with Mint and our custom front-end preinstalled. Our first version was built around Mint 7.
Right at the start of the project, I approached Clem Lefevbre and got his explicit written approval. We haven't modified Mint in any way except by installing it and adding and removing a few features, and there are
no source code changes to anything, so there is nothing for us to publish or disclose; all we are doing is installing the distribution, updating it and adding some freely-available drivers (e.g. the latest version of the HP LIP subsystem).
We do not distrubute the OS or the Envelope in installable form, we do not produce modified installation disks and the software is only available installed on our systems - it is not available for customers to install themselves.
Our custom front-end is written in Sun/Oracle Java and is not open source or Free Software; it is only available from us on a commercial, paid-for basis. The Envelope and its tutorials are what we are selling, rather than the OS underneath.
Although we do have formal written permission from Mint, we have not had any actual support or assistance or any other services from them, and we handle customer support ourselves. One of the terms of our agreement, with which we were happy to comply, was that we would clearly state in our advertising and promotional materials and so on, that we were using Mint. We've had a lot of press coverage over the last three years and it has resulted in good publicity for Mint - lots of free advertising! In interviews and so on, I and my colleagues have explained that people can download and try Mint for themselves and we actively encourage the friends and families of our users to do so, so that they can learn about Linux and perhaps provide assistance and help if it's wanted.
So, yes, it is legal and it has been done with the Mint project's knowledge and approval. If you had been curious, you could simply have approached us and asked! We'd have been happy to tell you all about it.
If there are any other questions, do please ask. Although I've been on here since 2009, I tend not to frequent any web fora much - but I am on the Linux Mint mailing list or you can email me directly. There's contact info in my profile.
http://www.simplicitycomputers.co.uk/