The most common and most likely answer to this crossword clue is the 8 plus 7 letter word AMERICAN ENGLISH
I've seen this crossword clue in The Sunday Telegraph
A
M
E
R
I
C
A
N
E
N
G
L
I
S
H
More clues leading to the result AMERICAN ENGLISH
Buddy explains!
The straight part
We know that "In which our conclusions are often non-U" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case in which our conclusions are often non-u = american english.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.