I looked into it very seriously.
It is certainly a viable option and depending on the work required and your own vanity when it comes to results, it is well worth looking into. By that I mean the services on offer in Hungary were far higher standard than those available through the NHS, and far cheaper than going private in this country.
In this country for cosmetic fillings and a good cosmetic crown at the front of my mouth (following a MTB face plant) I was looking at around £2k private in this country. In eastern Europe I was looking at £800 including a weeks stay in a hotel.
When it came down to it though, I managed to find a good local NHS dentist, who also did private work. She talked me through the process, explained that most private dentists are NHS trained any way, showed me the types of crowns available. (Basic cosmetic crowns are available on the NHS for damage to the front teeth, any non visible teeth get metal lump stuff). In the end I was satisfied that the work would be of good quality and cosmetically acceptable. The cost was the top tier NHS band which equates to £200.
So in summary, my advice, don't rule out the NHS. But by all means consider the tourist route.