Her birth index:

First name(s) DOREEN WEVIA
Last name DE COURCY-HAMILTON
Birth year 1905
Birth quarter 4
District Southwark
County London
Country England
Volume 1D
Page 16
Record set England & Wales Births 1837-2006
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These are births on FreeBMD and FindMyPast for Normans with Hamilton mothers and between marriage and Doreen reaching 43:

Surname First name(s) Mother District Vol Page
Births Mar 1936   (>99%)
NORMAN David H    Hamilton Marylebone 1a 755
Births Dec 1939   (>99%)
NORMAN Caroline J Hamilton Holsworthy 5b 1161
Births Jun 1941   (>99%)
Norman Eleanor M    Hamilton Islington 1b 111

It will need a birth certificates to find if any were theirs.  Start with the first?
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However The Times only shows this announcement:

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

This is very likely to be the above Caroline J Norman’s birth.  So get David H’s certificate first?  Ordered on 23 Feb 2017.

And David Hilary was indeed Doreen’s son so is to be entered up and followed up.
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Her household in the 1939 Register:

The Old House Yeolmbridge Werrington Park , Broadwoodwidger R.D., Devon, England
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FIRST NAME(S) LAST NAME(S) DOB SEX OCCUPATION MARITAL STATUS  SCHEDULE  SCHEDULE SUB NUMBER

Doreen De C Norman 21 Nov 1906 Female Private Means  Married  25  1
Constance Luckes 06 Jun 1915 Female Children Nurse  Single  25  3
Rona E Pethick (Rowe) 21 Jun 1917 Female General Servant (Domest  Single  25  4
Lucy Norman 23 Aug 1875 Female Private Means Widowed  25  5
Ruby Wakely 02 Oct 1914 Female Children Nurse Single  25  8

Note first that the year of her birth is different by a year to that on her death rindex, below.  As her birth index also has her born in 1905, I would suggest that the enumerator of the 1939 Register made a rare error.

Note second that her husband was, naturally absent, probably at sea.

Note third that there is a Lucy Norman in the house, a widow of private means.  My guess is that she is Doreen’s mother-in-law and thus could provide and identifier for his family in the census, with Lucy’s birth date.  This enabled me to find lUcy and an unknown son called ‘Master Norman’ in some Naval establishment in the 1911 census.
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Her death index:

First name(s) DOREEN WEVIA
Last name DE COURCY NORMAN
Gender Female
Birth day 21
Birth month 11
Birth year 1905
Death quarter 3
Death year 1997
District Taunton
District number 7241A
Register number A59B
Entry number 265
Date of registration mm/yy 0797
County Somerset
Country England
Record set England & Wales Deaths 1837-2007
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Her probate index:

Surname First name Date of probate Probate number  Date of death  Document type  Registry
NORMAN DOREEN WEVIA DE COURCY 08 December 1997 9780610293  07 July 1997  Grant and will  Bristol

Her will has been ordered on 7 Mar 2017 to see if it throws up any grandchildren.
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Her will arrived on 9th March.  The will was dated 7th April 1993 and its one codicil was dated 23 January 1996.

The beneficiaries were:

1. Her daughter Caroline Judith Beagley,
2. Her husband Hilary Norman, who was definitely still alive in 1993 and possibly so also in 1996,
3. The children of her late son David Hilary Norman.
4. The children of her daughter:
   (a) Alexander James Beagley
   (b) Emma Clara Beagley
5. Her daughter-in-law Diane Norman (presumablt the widow of David Hilary Norman)
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No Consular or military record found for their marriage.