What are the most commonly spoken languages of the Himalayas?

It entirely depends on the country.

In India/Pakistan: Hindi, Garhwali, Kumaoni, Urdu, Sanskrit

In Nepal: Nepali, Bhutia, Lepcha, Limbu, Newari, Kulung, Gurung, Sherpa, Tamang, Sunwar

In Bhutan: Dzongkha, Dzala

In China: Mandarin, Tibetan