The earliest type of prenatal test where you can do genetic testing of a fetus would be the CVS, which can be done between 10 and 13 weeks of pregnancy. After that, the next available option would be amniocentesis, which is usually done between 15 and 20 weeks of pregnancy. Both of these procedures do have some risk of miscarriage, so they are not typically done unless there is a medical indication (family history of a genetic disease, mother’s age over 35 years, abnormal blood tests, etc).
What type of genetic testing are you referring to? Most of these tests do not require any blood sample or consent from the father. If you are talking about paternity testing, however, this can’t be done without a blood sample from the father.