Flying with a nyckelharpa is stressful. The best advice is to not make it any one’s problem. Have a soft case, hold the instrument low, even behind you. Eyeball and flight attendant, get on the flight fast and get it up in that overhead bin. If you need to book priority, to get on fast, do it! I’ll add different airlines below as I investigate their policies.

Easyjet seem to be musical instrument friendly. Make sure you print their page off to prove you’re allowed!
From their website: “If you’ve booked a Standard seat or haven’t selected a seat, and want to bring on board a musical instrument that is larger than 45 x 36 x 20 cm (but smaller than 30 x 117 x 38 cm) you will need to book an Up front or Extra legroom seat in order to get a large cabin bag allowance.”