Not illegal for a dentist to work without a nurse, they may in n emergency.
But this is difficult and slow for the dentist and less comfortable and occasionally risky for the patients. Higher chance of lip burns from shank of slow-speed drill, tongue cuts from high-speed drill and what if a medical emergency occurs?!!! Usually means poor infection control unless that dentist is willing to run late because they are cleaning up properly between patients. What does your infection control lead say?
The higher number of patients a dentist can attend to in a day with a nurse compared to without a nurse means phoning the agency for a temp for the day effectively costs them nothing. (A day's wage for a dental nurse is hilariously small compared to other costs). The only reason I can think of for this happening apart from sickness is if there are a lot of dentists/surgeries but gappy diaries.
Reception need to organise the diaries better in the long-term. They may not know this if nobody points this out. They my think everything's just fine.
If you think you things are not improving you could put your CV out there and consider your options .
Call a meeting with the registered manager and ask : how long do you think I can keep this up? They are being horrible to patients, dentists and YOU. They are not making wise business decisions. Is the manager prepared to step in him/herself in case of emergency absence of a dental nurse? I think the trend to hire practice managers from retail backgrounds or with a business management degree is bad news, probably think we are just there to look pretty. If your manager is one of these they would benefit from putting their name on a dental nurse training course and stepping into the surgery.