As far as I know, the EU does not test for M. genitalium routinely. They actually follow similar guidelines to the UK, only testing people with specific symptoms (like unexplained urethritis or PID) or those whose partners tested positive. It's not part of standard STI panels there either.
Also, about HPV testing, the UK does use HPV testing as the primary method in cervical screening for women/people with a cervix aged 25-64. The US approach is pretty similar, especially for those over 30. Neither country includes it in general STI panels, or should, unless specifically needed, i.e, a high risk patient.
From what I understand, the reasoning is similar everywhere: For M. genitalium, many infections don't cause symptoms and might clear up on their own. Testing everyone could lead to unnecessary antibiotics and resistance. And for HPV, it's super common and often temporary, so they focus testing on high-risk types that could lead to cancer.
So the approaches between countries aren't as different as it might seem! Though I should mention that in my experience with the NHS, guidance can sometimes vary clinic to clinic because of misinformed staff who don't always provide the correct policy details. I've encountered this confusion several times myself in the past.