The most common and most likely answer to this crossword clue is the 8 plus 7 letter word STANDARD ENGLISH
I've seen this crossword clue in The Times
S
T
A
N
D
A
R
D
E
N
G
L
I
S
H
More clues leading to the result STANDARD ENGLISH
Buddy explains!
The straight part
We know that "that many of us speak" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case that many of us speak = standard 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.