Please tell me what will doctors do to help save someone who attempted suicide by cutting their wrist?

You can be admitted : You can be admitted into a psychiatric ward of a hospital or inpatient setting for observation. These would include psychiatric , psychological testing. And quite possible a medication stabilizing regimen. As well ongoing appointment with counselors, doctors' etc.

Cutting : The full psychiatric evaluation is medically indicated, immediately. Chances are the person will have to be inpatient for a while until self harm is no longer an issue. Follow up psychiatric help with medication and intensive psychotherapy should be the next course of action for this person.

If you are in danger of hurting yourself the doctor has a duty to warn to make sure you are in a safe environment. This might be a hospital. Please have a physical exam to know if these symptoms are due to an illness. If not see a Clinical Psychological with expertise in multiple types of psychotherapy and a Psychiatrist for medication.