As outlined by Major Shane Burton (RGR), Chief of Staff at the Headquarters Brigade of Gurkhas:

“The Brigade of Gurkhas is an integral part of the British Army and as such these troops are indeed part of Nato.

Gurkha units have been deployed as part of British Armed Forces supporting numerous Nato operations including Bosnia, Kosovo and Afghanistan.” (Soldier, 2019, p.55).

Reference

Soldier. (2019) Are Gurkhas Part of Nato Picture? Soldier: The Magazine of the British Army. August 2019, pp.55.