This is definitely a tricky one. I also originally voted for mouth cancer, influenced by the fact that even if a head and neck cancer doesn't originate in the mouth, the treatment is likely to affect it in some way. Also, as Brenda has pointed out, all of Dr Joshi's hard work has revolved around the 'Mouth Cancer' foundation and so changing names at this point doesn't seem very feasible.
However, thinking about it, nobody at the Maxillofacial unit where I had surgery referred to mouth cancer, it was always head and neck cancer, which is much less of a mouthful - excuse the pun - than oral, head and neck cancer and I think quite easy to grasp. In fact the job title of my Macmillan nurse was Head and Neck cancer specialist not Mouth cancer specialist and all of the blurb I was given to read, like stuff from Cancer Backup, used the term head and neck cancer.
In conclusion, I am in favour of using the term mouth cancer to attract attention to the awareness campaign, as long as all cancers of the head and neck are then included in the campaign material. Hope that makes some sense.