Usually yes on both, so sort them before your first class. Most instructors carry public liability insurance, and many studios want to see it before handing over the keys. If you're regularly
running drop-in fitness classes, it's worth keeping that cover and your instructor qualifications current, since insurers and venues often ask to see them.
Music is the part people forget. If you play recorded tracks in your sessions, you'll usually need TheMusicLicence.
PPL PRS explains that gyms, fitness, and dance studios playing recorded music generally need it, and
ukactive confirms the same obligation for fitness businesses.
Pro tip: Booking a private indoor Peerspace studio skips the outdoor training permits that London parks and councils require, so you avoid that whole application just to run a session. Check with your host whether music played in the space is already covered.