I am chasing a solution to this on a daily basis. NHS England have committed to commissioning a temporary local solution for NHS patients using existing dental surgeries in Kirkbymoorside and Pickering. These services should be up and running by the end of October but I understand that this may be delayed by a legal challenge from the previous operator.
NHS England are also looking to commission a permanent replacement as soon as possible. I have been approached by an experienced North Yorkshire provider, but frustratingly, procurement rules will mean that they have to go through a formal bidding process that may take 12 months or more.
In the meantime, if you need emergency dental treatment you should contact the NHS at https://111.nhs.uk/