It all depends on your insurance. If you cannot self-refer, I would certainly be insistent with your Primary Care doctor about seeing a Pulmonologist.
Regarding Mayo, they can move things along when they think it is the case. I sent my CT and PET scans (they want the actual scans not the report so be sure your doctor puts in the request to do that) and I was called the next morning by the Mayo Pulmonologist.
My x-ray said wait 6 weeks to do another x-ray and then CT scan, but I had local Pulmonologist order CT scan immediately. I read those reports and looked my local Pulmonologist in the eye and said I was going to Mayo. They were not happy, but I have original Medicare and pay mucho money for my Medigap insurance. I learned that it is very important to understand exactly what type of treatments are available and if your clinic/provider is capable of providing all options. My local clinic could not do VAT surgery for example. Also, the local Pulmonologist wanted to do a needle biopsy on the largest area, but their Radiologist refused to do it. He said that it was too risky, and he could not get enough sample tissue.
Mayo offered me surgery immediately and said 80% chance cancer. They did bronchial biopsy and VAT surgery together. Also, the first local CT scan only picked up two areas of concern. Mayo CT scan prep for surgery identified other nodules – they set their machines to do this. So, I was diagnosed with multifocal adenocarcinoma. They took out the largest area which was cancer and now I sit back and wait to see what the other nodules do. The area in the left lung which also lit up on the PET scan has not changed in 10 months and could still be scarring and not cancer. The other nodules in the right lung are part solid and ground glass. They don't want to do needle biopsy.