First Officer reported Captain flew an unstable approach resulting in GPWS alert and late go around to a landing.

Narrative

On Date at approximately XA:50 while working a trip as the First Officer and Pilot Monitoring from ZZZ to HPN; The Captain had an unstable approach going full deflection on the glide slope. During the approach into HPN; we were in IMC conditions. We were given the ILS 16. ATC vectored us onto the approach where the Captain armed the approach for an ILS 16. Continuing inbound the auto pilot did not capture the approach at that time. The Captain noticed the ILS was not captured; the LOC nor Glide Slope. I asked The Captain if we should perform a go around; he told me no. He tried to correct it by pressing V/S and manually putting our required descent rate in. This did not seem to work so the Captain discontinued the autopilot. The Captain proceeded to disconnect the autopilot in order to hand fly the aircraft to get it re-established on the approach. As he continued we ended up going full deflection on the glide slope. At approximately 750ft MSL and 2NM from 16 runways threshold. Tower advised we were not on the LOC and asked us if we had the correct altimeter in; which we did. Ceilings were at 400ft and visibility was around 1600ft. The tower advised we were low and we needed to go around. I advised the tower we were going around. As we initiated the go around we got a GPWS warning. We were given an assigned heading and altitude during the go around. The Captain may have exceeded airspeed limits in the D airspace going approximately 250knots before slowing. We performed the go around safely and completed checklists. We were able to get vectored back around to shoot the approach again. We loaded the ILS 16 and shot the approach for the second time. We loaded the approach which then activated properly following the localizer and glide slope. We were able to get out of IMC conditions before minimums and had sight of the Runway. The Captain disconnected the autopilot and landed the aircraft safely at HPN.