does a psychologists or psychiatrists automatically have the authority to diagnose ADHD?

When you qualify as a clinical psychologist or psychiatrist, you will have studied ADHD and the criteria necessary for such a diagnosis. There wouldn't be a separate training course on first qualifying although Continuing Professional Development might cover it in more detail.

I have an ADHD diagnosis BTW.

A therapist /psychologist can diagnose a patient as far as i know.  But they aren't able to prescribe medications so that would be up to a psychiatrist or other doctor to handle.  

In the US, state laws for licensing of psychologists vary; in most states, psychologists can diagnose ADHD and other mental health disorders, but cannot prescribe medication.

They don't here in the UK. My last psychiatrist worked for an adult ADHD assessment team in his prior job so he was highly experienced in diagnosing ADHD. At one appointment, after getting to know me for several months, he told me that I had ADHD, gave me a print out on medication, and told me to go away and research it. He got in trouble at the next team meeting as only ADHD assessment teams are allowed to diagnose people and he had overstepped his role. I have a private diagnosis so it doesn't matter but if you're looking for a diagnosis you'll need your GP to refer you to an assessment team, if one actually exists in your area. Waiting lists are ridiculous as well with most people waiting over 3 years to be seen. It's different for children or adolescents but I'm not familiar with the process.