How did the BAC become permissible in court or is it strictly based on their suspicion?

I was discovered by police in my vehicle sleeping in the parking lot of a grocery store after they responded to a call about a suspicious subject. I had been driving from a bar in Michigan and had stopped in the parking lot to rest and sleep off any intoxication and it was also early morning. After tapping on my window and making contact with me, the officers kept reinforcing they were checking on my well-being, thanked me for cooperating and did not acknowledge that I had been drinking or any mention that I was under arrest. They did put in their report later that there was vomit on the seat. I then agreed after they coerced me to go to a hospital. The officers waited in the lobby and I agreed with a nurse to do blood draw under the agreement it was not being ordered by the officers and they wouldn't see the results. Approximately five minutes after the blood draw, I was moved from the bed, placed in handcuffs and thrown up against a wall and transported to jail mostly without any interaction from the officer. Nearly two months passed and I received a charge of OWI under the state's super drunk law. My question is, did I have any protection from them receiving the results of that test without me permitting since I was not under arrest until after the test? Is it typical in DUI cases that a warrant be filed for blood tests which causes the 2 month delay? How did the BAC become permissible in court or is it strictly based on their suspicion? Thank you.