No, The Royal Marines Band Service is a separate entity.
They undergo limited military training but do not complete the Commando Course as such, as far as I am aware.
The Army is(or at least was) slightly different. When I did infantry training, those going to become bandsmen still had to complete the full 18 weeks training, not just the 6 week CMS(Common Military Syllabus).
They then went on to whichever School of Music they were designated for. Drummers however are recruited from within the ranks of an infantry unit and a Drums Platoon would specialise in support weaponry in a combat role i.e. GPMG in the Sustained Fire role or similar.
From Mamya's link:
In addition to music making, Royal Marines Musicians and Buglers are fully trained and operational military personnel and usually perform support duties such as logistics and medical personnel and drivers. Additionally they may also provide entertainment for troops deployed overseas and represent the Royal Marines in an ambassadorial role.