Maternity insurance

I understand that we are able to get maternity insurance from week 13 onwards. N I’m turning 11 weeks tomorrow and on week 12 I’ve an appt with the gynae to do an Oscar test to check for Down syndrome. N the insurance company tell me it’s not advisable to do the checks before week 13 in case there is some medical issues, I might have difficulty getting insurance coverage. What should I do? Does Oscar test result come out on the same day? And also…I’m currently also on meds for my depression n anxiety, seeing a dr for it. Does that means also that I might have difficulty getting insurance coverage?