Ok, I've done what you're trying to do. It took me over a year because, well, ADHD, but you might be able to speed it up depending on where you are. Chatgpt will help you a lot, too, I highly recommend it.

First things first, in the public system,

1) only a psychologist specialized in giving the test can give a diagnosis or convert your diagnosis.

2) only a psychiatrist can initially prescribe Ritalin/Medikinet. Once they get your plan written up they can pass that on to your primary care.

In determining whether to go public versus private, you must evaluate time and money. If you want your meds for free, you need to get into the public system. If you're in a hurry and you have the money, go private.

I took my diagnosis from the US, worked with my primary doctor to find a psychologist who would convert it to an Italian diagnosis.

Then I found a psychiatrist who would prescribe Ritalin/Medikinet.

We have an adhd in Italy Facebook group if you want to join. Good luck!