Her birth index:

First name(s) CLAUDINE M I
Last name WYMER
Birth year 1961
Birth quarter 3
Mother's last name Leachman
District Witney
County Oxfordshire
Country England
Volume 6B
Page 1698
Record set England & Wales Births 1837-2006
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There are 7 births of Siths with a Wymer maiden name for the mother.  Of these 2 are in Durham, 2 in Oxford and 3 in Norwich.  I wonder which are the family of Claudine and Christopher?  A first step might be to get one of the Oxford birth certificates.
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In 2005-14 the electoral registers show here to be living with, in various years:

  Michael Whitty, Damian Smith, Geoffrey Smith, Patricia Wymer, Christopher Regent

Of these Patricia Wymer is definitely her mother, Christopher Regent could be her brother-in-law, Damian and Geoffrey Smith probably the Oxford pair born Damian Edward T and Geoffrey Raymond in 1983 and 1981 respectively.  Michael Whitty is anyone’s guess.

So it looks highly likely that her and Christopher Smith’s sons are Geoffrey and Damian.  Let’s get Geoffrey’s birth certificate, index details as follows:

First name(s) GEOFFREY RAYMOND
Last name SMITH
Birth year 1981
Birth quarter 3
Mother's last name Wymer
District Oxford
County Oxfordshire
Country England
Volume 20
Page 3038
Record set England & Wales Births 1837-2006

Note that this will be a phone jobbie as it is within the last 50 years.
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Their marriage index:

First name(s) CLAUDINE M I
Last name WYMER
Marriage quarter 3
Marriage year 1980
MarriageFinder™ CLAUDINE M I WYMER married
Christopher M Smith
Spouse's last name Smith
District Witney
County Oxfordshire
Country England
Volume 20
Page 3657
Record set England & Wales Marriages 1837-2008
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