Does "nonstandard English" come across as judgmental in the following context?

Use of "nonstandard accents" implies there is a "standard accent".
Is there? Which is it? Does most everybody agree on it being the standard?

If the answer to any of those questions is "no", and if you care about not offending other people (however irrelevant the offending point might seem), then I agree that alternatives like "certain accents" or maybe "unfamiliar accents" are more neutral, and so would serve a better purpose in establishing that rapport you very appropriately talked about in your answer.

Just for the record: today, saying that there is such a thing as a "Spanish standard accent" would be fairly offensive to 90% of native Spanish speakers (and there's half a billion of us).
Sure, some accents are understood better by some people, some accents are seen as more "intellectual" etc. -- but to put one above the other would be disrespectful. There is a fair agreement over all Spanish academies, that every Spanish variant is as Spanish as any other.

I'm sure any Mexican would be as mad if I told them their accent was "nonstandard" as I would if they told me the same.
I can only guess that the same applies to English speakers.

Excellent question, because it is hits a lot of issues with disparagement, taboo, euphemism (and its treadmill).

Sure, any word has its natural strength of elicited emotional response. 'Grass' is pretty neutral, 'fat' some could argue is descriptively neutral but nobody likes to be heard called that, 'bastard' seems always disparaging even if it has a non-judgmental literal meaning, 'shit' will have your grandmother annoyed or actively tell you to watch your language, and there are further words that will have her slap you.

There are a number of words to describe how... hm... how to put this tactfully... how much of a member of ... the ... no... a group. No judgment on those in or out of the group. Just inside or outside. There's 'minority' (If in fact that group is smaller. There's 'normal' but that seems judgmental. 'Average' is so clinical, yet has its own problems. Neurologists use neuro-typical and -atypical for people towards the tails of behavioral spectra (eg autism).

The British use 'received' (as in 'Received Pronunciation") as some radically misdirection by metonymy or implied absence or ... I don't know what. What's being received? Who is giving and who taking? Never mind, I don't think that's really part of it.

Anyway, all these alternatives for someone who doesn't speak like you could be problematic simply because they point out a difference, and the concept of the difference is a problem. And to different extents, the messenger, the word that is used to label the difference, can be blamed or accrue varying degrees of the emotional baggage of the concept.

For pronunciation (and other aspects of language), the first descriptor people use is 'right/wrong' or 'correct/incorrect'. For example, "It is incorrect to pronounce 'wash' as 'warsh'". This is how we think school teachers tell us how to conform to a particular variety. It has the force of such schoolteachers, which usually translates to shaming. And even on sites like ELU there is a slight tendency to fall back on that elementary and tendentious language.

But the alternatives are like those above. Normal? Average? Minority might work but has its connotation problems and also may not be literally true. Informal? That's often the case and is not terribly judgmental, but may not be actually in fact true.

The word usually used to describe this is 'non-standard'. 'Standard' is what people are 'expected' to speak. It is the most euphemistic so far. There may be others, but this is the word to use. I think I'd be pretty upset if someone told me I speak atypically. Even if it is true. That's pretty clinical. If it comes across as judgmental... maybe it's the messenger, or maybe it's the message.