Nurse here. Nurses can't order tests like this without an MD/DO/PA/NP. Though as you said, nurses can make a suggestion whether or not to run a test, and the provider can then sign off on it.

Since the nurse is bedside and the provider is rarely up-to-date by the minute, it's common practice that the nurse make judgments on a patient's immediate needs as their condition changes, and pass a suggestion to the provider.

These are usually reserved for high acuity scenarios where a patient's condition is actively worsening and the nurse needs to do something right then and there (i.e. "Patient's BP is dropping, becoming lethargic, and the peri-pad is saturated x2. Would you like me to bolus NS and order a blood transfusion?").

Ultimately, though, it's the provider's decision whether or not to act on these judgments and make the orders. I'm not an L&D nurse so I'm not sure if you need signed consent for that particular test, but if you do, then the provider will need the consent form before ordering the test.