B737 First Officer reported distraction prevented noticing Pilot Flying descended below path; resulted in ATC low altitude alert and GPWS terrain alert.

Narrative

We were on approach to RWY 30 into BZN on the RNAV RNP 30 from Livingston transition. We were getting moderate turbulence with the wind around 600-70 kts below 10;000 [feet]. We had thoroughly briefed the approach prior to the descent. While on the approach around 5;000ft Tower advised us the an aircraft had circled to 30 from ILS 12 and during the approach experienced a 2;000 fpm updraft and moderate turbulence on approach but braking action was good upon landing. Around LAVPE; the Captain called for below the line landing checklist while we were descending on the VPATH. I ran the checklist and then looked at my Jepp Approach Chart to confirm what the missed approach altitude was; it was right after that we got a 'Caution Terrain' aural warning followed by Tower telling us we had a low altitude warning. We then got another caution terrain warning and the Captain disconnected the AP and started to climb; I saw at that point we were below the VPATH somehow. We started to climb back towards the path and then got the RWY in sight and descended for a normal landing. Some time between me going heads down to run the below the line landing checklist and MOSNE intersection we got low and I still do not know how. We were on AP and VPATH just a few seconds prior; I don't know if we got a downdraft or what but I'm still confused as to how the aircraft got low. We decided to continue to the approach instead of going around and I believe that was the safer of the two options giving the wind gusts and moderate turbulence in mountainous and unfamiliar (for me first time to BZN) terrain.We both had a long day (I commuted in that morning because I missed my last commute the night prior due to operational delays) and had been fighting turbulence all day long on ZZZ-ZZZ1-ZZZ-BZN routes which probably made us more tired than normal. This was my first time into BZN and the low visibility; moderate icing from the snow storm; and moderate (or greater) turbulence on approach made it tougher to concentrate and multi task.I'm not sure how we got below the path on approach as I was Pilot Monitoring and heads down running the checklist so I have no suggestions as of now. I should have done a better job monitoring our descent and VPATH even while conducting the Landing Checklist.