No. Neither Alan Watts nor Bruce Lee were Taoists. Alan Watts was a Western thinker who read a lot of Eastern material, but never dove into it experientially. I do not criticize him, because genuine contact with Eastern wisdom was very rare in his time.
Bruce Lee was a martial artist who’s style, Win Chun Kung Fu, came through a tradition of monastic training in Ch’an, Chinese Zen, as well as other sources, and there is some Taoist influence there.
If Bruce Lee had encountered a Taoist sage and holder of qigong wisdom at the right time, he might not have died so young. Qigong training teaches us to hold the yang power of the martial arts in a larger framework of yin energy, which is healing. Lee, instead, generated tremendous yang fighting energy, and, when injured, had no recourse to Taoist methods of healing. The Western methods used to relieve his pain only masked the problem and prevented the natural flow of the healing Tao.
Fortunately, more authentic contact with the Tao is possible now, in our generation. Unfortunately, there is a lot of misguided stuff out there, as well.
Alan Watts described himself as a Hindu/ Taoist, Buddhist. He was a great scholar and also very intellectual and a great educator and entertainer at the same time. I am not sure if he practiced any of the philosophies he taught about that much. However I am lead to believe that he did practice Tai Chi. So, you could describe him as a Taoist if you like. Perhaps he was more of an Orientalist.
Bruce Lee was not so religious, I think. However he was very philosophical and his Martial Arts skills were a mix of Taoist and Buddhist Kung Fu and other styles. The way he spoke suggests that he had a deep connection to Chinese philosophy in general.
I think that Alan Watts was definitely a follower to a certain extent of the religions he taught about. With regards to Bruce Lee, I think you would have to find out from his history and the people who knew him. If Bruce Lee would have lived into his middle age and old age then I think that he would have become more spiritual and maybe he would have practiced Taoism and/ or Chan Buddhism which is a mixture of Taoism and Buddhism. Many Cantonese Chinese, especially in Hong Kong practice Taoism and / or Buddhism.
