When the NHS first came in, dental was included but it became very quickly obvious that it was costing an absolute fortune*, so they stopped it. 

*The nation actually had really bad teeth then. Until pretty recently we had very healthy but ugly teeth, that may be changing now that there are so few NHS (subsidised) dentists around. From 2016:

In the study, published in the British Medical Journal, researchers in the UK and the US examined data from English and American dental surveys.
The results showed that the average number of missing teeth was 6.97 for English participants, but 7.31 for those in the US.
Additionally, people were more likely to suffer poor dental health in the US because of socio-economic factors.

https://dentistry.co.uk/2016/01/06/english-have-better-teeth-than-americans/ 

 Dental really needs to be covered under any medical scheme because things like gum disease eventually cause heart disease.  You can't separate out body parts and pretend they don't function as a whole.