The most common and most likely answer to this crossword clue is the 5 plus 5 letter word BILLY GOATS
I've seen this crossword clue in Financial Times
B
I
L
L
Y
G
O
A
T
S
Buddy explains!
The straight part
We know that "They’re responsible for the kids, but not without a nanny" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case they’re responsible for the kids, but not without a nanny = billy goats.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.