B737 pilot reported receiving a terrain alert while on final approach; resulting in a missed approach and diversion.

Narrative

On a turbulent descent and arrival into SFO; due to weather in the area; we were given the RNAV approach to [Runway] 10L. This was after we had been vectored due to airport saturation. We set up for the approach and briefed it. The missed was briefed and it was 4;000 ft. We were vectored onto the approach just outside of the FAP; while we were turning onto the inbound leg the autopilot disconnected due to what I expect was turbulence; and would no re-engage. The Captain hand flew the rest of the approach. When we were approaching the MDA; approximately 200 ft. above; we got a TERRAIN; TERRAIN alert and I immediately called for a go around. The go around was performed perfectly except Tower told us to climb and maintain 3;000. I never adjusted the MCP as we were busy with the missed procedure. Tower queried us about the altitude and said to just maintain 4;000. They then asked us about retrying the approach; we decided as everyone seemed to be going missed; to divert to ZZZ. The flight landed uneventfully at ZZZ.