The most common and most likely answer to this crossword clue is the 11 letter word ALDERMASTON
I've seen this crossword clue in The Guardian
A
L
D
E
R
M
A
S
T
O
N
Buddy explains!
The straight part
We know that "the end of March in 1958 /the end of March in 1958" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case the end of march in 1958 /the end of march in 1958 = aldermaston.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.