Charter flight crew reported malfunction of aircraft flight guidance system causing terrain warning on approach.

Narrative

While flying from ZZZ1 to ZZZ; Captain and I were vectored and cleared for the ILS XX approach into ZZZ. I was PF; Captain was PM. We were established on the approach; configured; and had received landing clearance from Tower. As we began our descent out of 1600ft; we entered a hard IMC layer. Everything was on speed; on path. Autopilot and auto throttle engaged. Between 1300-1000ft; we lost Flight Guidance Computer 1 (as we later realized); and the aircraft pitched HARD down; into a 2700-3000fpm uncommanded dive. Upon feeling the abrupt pitch down; I reached for the disconnect switches on both AP (Auto Pilot) and AT (Auto Throttle); as the systems disconnected; we got alerts of 'Terrain Terrain; Pull up'; as well as a call from Tower to 'check altitude and climb immediately'. I pushed full power and pitched up to escape the situation; then began the missed approach. Later; upon landing; Tower reported that our altitude showed 900ft when she gave the call. We executed the missed approach; returned to ZZZ on Flight Guidance Computer 2; and landed without further incident.