Even native speakers can have trouble understanding some British accents. Most of my family has lived in Texas for several generations and are monolingual English speakers, but one cousin was born in and has lived his whole life near Liverpool, and sounds a bit like it (omitting long story about immediate aftermath of WW2 and my feckless great uncle). His wife is from Cheshire and has a rather strong accent. I understand her just fine, but went to college on the East Coast, had professors and classmates of varied linguistic backgrounds, and have spent most of my adult life in Germany.

When they visited Texas for a family reunion, several of my cousins who have not spent much time around non-Texans honestly asked about the language she was speaking.

More of a tangent as I agree with your point, but...

I think this diversity, while large, is still less than in many other languages. As one example of many, just within Europe there are accents/dialects of German in Switzerland that are largely mutually unintelligible even to some other Swiss German speakers, and nearly entirely unintelligible to most Germans. This is also true for dialects of Italian, or even Dutch, according to friends from those countries. It's generally even more fragmented outside of European languages.

Unlike English, some of these cases don't even have easily standardized registers. In my experience, people in Birmingham or Glasgow, for instance, can fluidly switch to a more standardized accent and dialect that's natively or "intuitively" understandable to a speaker from North America or South Africa.

A few years ago I visited an office in Birmingham which is less than 100 miles. Someone was trying to tell me a password and I had to get them to write it down after several failed attempts. The basic sound of some of the letters was just completely different. I also have a strong accent which can be much harder to understand than a none native speaker.

I'm not the person you asked but I am someone who worked to change my accent (successfully enough that I have gone from a thick, easily identifiable accent to not having anyone guess that I was not local to where I live now in the last 15 years or so).

There's no real trick to it other than paying attention. Record yourself speaking, listen closely to how the sounds you use differ from the accent you want to have. Then practice saying those sounds over and over until they become natural. Keep at this for several years.

It's painful to do (though I find it less so if you focus mostly on specific sounds rather than listening to your full speech -- it's easier to separate the sound from the revulsion of hearing your own voice that way) but it works.