When researching this page we found lots of information from 2008 based around the old processes and old telephone numbers for NHS Direct.
After speaking to a few dental practices around town it seems the new process at the time of writing is to ring NHS Direct Wales on 0845 46 47 , that is the full number just 0845 46 47!
On dialing that number you will get given a few different options, and hopefully provided with the information on which dentists are open and are closest to you.
Its worth noting that there are definitions of what an ‘emergency’ is , namely unstopable dental bleeding, infections that are spreading and damage to teeth or the like due to an accident/incident.
NHS Direct Wales are pretty clear in saying that tooth related pain, however serious , is not classed as an emergency.