I have worked in 3 different countries (NZ, Australia, USA) and I have to say, in terms of needing orders from doctors, there really isn't much of a difference, if I had to quantify the number of times I had to chase a doctor for an order.
In both Australia and NZ, nurses can make use of medication standing orders for commonly used OTC drugs. Say for example you need paracetamol (acetaminophen) for a patient but it is not prescribed, you can initiate a once-only order (without a doctor's approval) for that medication as long as your patient meets all criteria. In NZ, I worked in an ER setting and I could basically obtain blood specimens and order blood tests on my own based on the patient's presenting complaints.
In the U.S., there PowerPlans. These are order sets where you can choose which orders to initiate based on your patient's needs. For example, your patient complains of a dry nose, so you can select normal saline nasal spray and initiate that order based on your patient's condition. So technically, it's already been approved by a doctor for that specific patient and all you need to do is initiate it (after making sure your patient does not have allergies to it and they meet criteria). There are certain blood tests that you can order for a patient too but they have to be again, part of a PowerPlan. An example of this would be the Electrolyte Replacement Protocol. If for example, your patient's potassium levels fall below normal, you can initiate potassium replacement and order a basic metabolic panel test for the following day.
The U.S. Healthcare system is so diverse that working in different hospitals can be entirely different experiences. So what other hospitals allow nurses to do without a doctor's order, other hospitals don't. And vice-versa.
For most other things though, because of insurance and litigation purposes, a doctor's order is warranted.