From The Times:

NORMAN--On Nov 25, 1939, at The Old House Yeolmbridge, Devon, to Doreen, wife of Lt-Commander Hilary Norman, R.N.--a daughter.
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Her birth index:

First name(s) CAROLINE J
Last name NORMAN
Birth year 1939
Birth quarter 4
Mother's last name Hamilton
District Holsworthy
County Devon
Country England
Volume 5B
Page 1161
Record set England & Wales Births 1837-2006

Her second nameis known from her mother’s will.
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There are two marriages on FindMyPast’s index:

Last name First name Born Died Event Location

Norman Caroline J — — 1974 Kensington, London
Norman Caroline J — — 1986 Ploughley, Oxfordshire

I incline towards the former as I might have expected her to marry outside London but in Devon or Somerset where her mother lived.  So the next move could be to obtain that marriage certificate.

An alternative might be to get her mother’s will, in fact that has to be the right next move.

In her mother’s 1993 will, she names her daughter as Caroline Judith Beagley and give her children as:
   (a) Alexander James Beagley
   (b) Emma Clara Beagley

This led immediately to finding that it was indeed the first marriage, with John A Beagley.  And this led to discovering precisely the births of those two Beagley children to a mother with the Norman maiden name.
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Their marriage index:

First name(s) CAROLINE J
Last name NORMAN
Marriage quarter 3
Marriage year 1974
MarriageFinder™ CAROLINE J NORMAN married
John A Beagley
Spouse's last name Beagley
District Kensington
County London
Country England
Volume 13
Page 2061
Record set England & Wales Marriages 1837-2008
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