The most common and most likely answer to this crossword clue is the 4 plus 2 plus 3 plus 4 letter word TURN OF THE YEAR
I've seen this crossword clue in The Times
T
U
R
N
O
F
T
H
E
Y
E
A
R
Buddy explains!
The straight part
We know that "when January begins" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case when january begins = turn of the year.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.