I thought I'd have a quick look at the GRO to see how many Leslie Bradford births were registered, and there are a few.
The 1939 is well known for inaccuracies (errors?) with birth years, at least with my family. I sometimes think they forgot the year they'd been born.
But they are often around the right date with the actual birthday, so the quarter of registration for YOUR Leslie is probably Apr/May/Jun (shown as J in the GRO list)
BRADFORD, LESLIE BERNARD VICTOR PEABODY
GRO Reference: 1902 J Quarter in ASHBY DE LA ZOUCH Volume 07A Page 114
is one possibility.
Or a bit further off with the birth year:
BRADFORD, LESLIE LEWIS WOOLLAMS
GRO Reference: 1905 J Quarter in KING'S NORTON Volume 06C Page 519
If I were you, I'd have a squizz round at possible marriages Bradford/Peabody and Bradford/Woollams to see what turns up.....