It probably doesn't have anything to do with the connections, just make sure they're connected tightly and that there is no damage to the connectors and the wires themselves. I'm assuming you've tried multiple games, and tried removing all accessories and anything else non-essential, but if you haven't that could help you see if any components are causing problems (in most of these instances the console will display an error message though). You can't get into the system menu, right?
If you can't find any solution, the console has a 12 month warranty on it (at least it does in Canada), you may have to send it to Nintendo for repairs. After looking through the Nintendo Europe site, it seems the nintendo service center for the U.K. is:
Nintendo Service Centre
Codestorm House
Walton Road
Farlington
Hampshire
PO6 1TR
Telephone: 0870 6060 247
..but I have no idea how useful that is to you.