OAK Tower Controller reported traffic overshot altitude clearance resulted in a low altitude alert and a loss of separation due to ATC procedure resulted in a CFTT event.

Narrative

Ceilings reported at 019 by previous arrival. Three commercial aircraft on final to runway 12. The first aircraft on approach slowed to approximately 130kts. I advised the second aircraft; Aircraft Y; that they had a 40kt overtake on traffic 3.5 miles ahead. They slowed down to ensure proper spacing. The third aircraft; Aircraft Z; initiated communication with me while 3.18 miles behind Aircraft Y with a 20kt overtake. I told Aircraft Z about the overtake and asked if they had Aircraft Y in sight. They said they were already at final approach speed and that they were IMC. At 3 miles separation I instructed Aircraft Z to go around; fly runway heading; and continue the decent to 1;600ft in order to stay under the SFO final. The pilot read back the instructions correctly but descended to 1;200ft. I issued a low altitude alert to Aircraft Z and told him to climb to 1;600ft. I then climbed him higher once he was clear of the SFO final. Aircraft Z and Aircraft Y got within 2.6 miles and 400ft. Recommendation: With low ceilings on flight plan we have no outs if we lose separation on final. If we have a go around we have to force the aircraft to continue the decent to get below the SFO final. When an aircraft increase speed during a go around this often leads to a loss of separation with the preceding aircraft. Recommend a minimum of 4 miles separation on final to 12 when ceilings are reported by the tower to be below 3;000ft.