At a private dentist, you can ask to see a different dentist at the practice usually without any problem. If you live in the UK and would like to see a different dentist at a different practice on a private basis you can simply call them up and ask for an appointment. You do not have to stay at the same practice all your life and you could see any dentist in the country as often as you like, as long as it was on a private basis.
However, most NHS practices are very busy, work to a budget and it may be against their policy to allow patients to change dentists within the practice, but you can always ask. Also, it is not usually possible to move to an NHS dentist due to tight funding constraints within each individual practice. There may not be enough NHS places to care for you at another dental practice. So you could either become a private patient and have the service you want or it is worth checking prior to informing your current practice you want to leave.