When can police get a warrant for my blood?

The mechanics are the same as any other warrant. An officer prepares an affidavit setting out the facts supporting probable cause, presents it to a judge, and the judge decides whether to authorize the draw. Electronic and telephonic warrants have made this fast enough that the old dissipation-of-alcohol argument for skipping the warrant carries less weight than it once did.

What's in the affidavit matters enormously and often goes unexamined. It's a sworn document written under time pressure, and it can contain assertions that the body cam contradicts, facts stated more strongly than the officer's own report supports, or omissions of information that cut the other way.

If a warrant was obtained in your case, that affidavit is obtainable. Reading it against the rest of the record is one of the more productive things a defense attorney does in a serious DUI case.