Do you have urgent walk in center's where you live? It may not be a dentist that you need to see and they'll give you pain meds and antibiotics especially if you have an infection. If you do need to see a dentist then they can refer you from there to see an orthodontist at your local hospital, its worth a try and better than sitting in A/ESo I called a dentist just after 9 who gave me an emergency number was on hold for 40 minutes and the first thing she said when she answered is we have no appointments but I can give you painkiller advice and told me to call back tomorrow at 8:45 I told her what NHS 111 said when I answered the question and she told me not to contact my GP or the hospital till I've seen a dentist
Good luck I hope you get seen to as soon as possible