Medical test in Europe

In what countries will you be when your husband will need to get these tests done? 'Europe' is not uniform when it comes to healthcare systems and the rules and services available can differ quite drastically across countries.

https://travel.state.gov/content/travel/en/international-travel/before-you-go/your-health-abroad.html

You can find lists of doctors and hospitals in the country you are
visiting on the U.S. embassy and consulate websites, under the
“American Citizens Services” heading.

This is our "we are retired!" trip with no set itinerary. By the time he needs the first test (October) we will probably be in southern France or Spain.

E.g. Spain: https://es.usembassy.gov/u-s-citizen-services/doctors/

In order to help American citizens, who might not speak Spanish, the
Embassy maintains a list of English-speaking physicians in the Madrid
area. The Consulate General in Barcelona and the U.S. Consular Agents
throughout Spain can provide information concerning the availability
of health care in their respective regions. The Department of State
maintains information which may be of use should you experience a
medical problem while outside the U.S.

https://es.usembassy.gov/u-s-citizen-services/doctors/

Medical Services List (PDF 390K) has been compiled to assist
American citizens in Spain in the event medical services are required.
The Embassy is unable to assume any responsibility for the activities
of the organizations and individuals listed, but great care has been
exercised to include only those who are thought to be capable,
reliable and ethical. Additionally, it has not been possible to
include in this listing all of the capable physicians practicing in
Madrid. Users of this list should inquire in advance about hours of
operations and fees.

PSA is a blood test screening for prostate cancer. You don't have to fast for that test. For doctor it is important to know what was the value of the previous test.

If you have medical insurance, call your provider. Even if your policy does not cover medical expenses in Europe, their customer service department should be able to give you guidance as to how to get the tests and have the results emailed or texted to your physician at home. Most of the larger medical insurers have networks and preferred providers in Europe.

We were in Siena two years ago, and I saw notices on the doors of pharmacies advertising blood tests. The same may happen in other countries, but I know it is not the case in the U.K. Here you would need to see a doctor first.

Every country has its own system.

Actually there are pharmacies in the UK do offer some blood tests, plus it is possible to visit a phlebotomist in a private clinic without seeing a doctor. Results from a private PSA test typically take a few hours.

For doctor it is important to know what was the value of the previous
test.

This. Make sure you have this info before you travel. The PSA test is VERY important.

https://www.cancer.gov/types/prostate/psa-fact-sheet