The death registration in 2002 is extremely helpful as it provides a date of birth:
Name: Lily Fluck
Death Age: 89
Birth Date: 14 Mar 1913
Registration Date: Nov 2002
Registration district: Ross
Inferred County: Gloucestershire, Herefordshire
Register Number: 19B
District and Subdistrict: 5241
Entry Number: 235

and we are able to search the 1939 Register and find this household:

May Hill Kilcot Newent , Ross and Whitchurch R.D., Herefordshire, England
FIRST NAME(S) LAST NAME(S) DOB SEX OCCUPATION MARITAL STATUS SCHEDULE SCHEDULE SUB NUMBER
Rosanna Gardiner 07 Apr 1872 Female Domestic Duties Married 1 75
Leonard Gardiner 24 Jul 1907 Male Farm Labourer Single 2 75
Lily Gardiner 14 Mar 1913 Female Domestic Servant Single 3 75

We have this birth registration:
Births Jun 1913
Gardiner Lily Bosley Westbury S. 6a 515

and marriage of parents:
Year 1899
Month Aug
Day 7
Grooms Surname GARDINER
Grooms Forenames Albert
Grooms Age 33
Groom Condition bachelor
Grooms Occupation labourer
Grooms Residence Lea Bradley?
Grooms Fathers Surname GARDINER
Grooms Fathers Forenames Leonard
Grooms Fathers Occupation miner
Brides Surname BOSLEY
Brides Forenames Rose Hannah
Brides Age 23
Brides Condition spinster
Brides Occupation
Brides Residence Aston Ingham
Brides Fathers Surname BOSLEY
Brides Fathers Forenames Jeremiah
Brides Fathers Occupation gardener
Licence or Banns banns
Date of Banns
Signature or Mark Both sign
Witness 1 Thomas Bidmead
Witness 2 Amelia Gardiner
Other Witnesses
Officiating Minister J Wilkinson Storey vicar
Event Marriage
Memoranda
Notes
Register Reference P155 IN 1/9
Page Number 37
Parish Chapel Gorsley

Albert’s baptism can also be found on this site.

As for William H Fluck we have this possible:

Name: William Henry J Fluck
Death Age: 89
Birth Date: 28 Sep 1905
Registration Date: Sep 1994
Registration district: Gloucester
Inferred County: Gloucestershire
Register Number: B43
District and Subdistrict: 4831B
Entry Number: 127

and Probate Index provides a little more information which suggests a good match:
Name: William Henry Jesse Fluck
Death Date: 2 Sep 1994
Death Place: Little Gorsley Kilcot Newent Glos
Probate Date: 11 Nov 1994
Registry: Oxford

This is the 1939 Register Entry:
Townsend Household (2 People)
The Quan Bristol Road , Gloucester R.D., Gloucestershire, England
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FIRST NAME(S) LAST NAME(S) DOB SEX OCCUPATION MARITAL STATUS SCHEDULE SCHEDULE SUB NUMBER
Adelaide L Townsend 04 Oct 1876 Female Unpaid Domestic Duties Widowed 44 1
William H Fluck 28 Sep 1905 Male Raf Transport Driver Single 44 2

His birth registration from GRO online Index:

FLUCK, WILLIAM HENRY JESSE WOOD
GRO Reference: 1905 D Quarter in WHEATENHURST Volume 06A Page 337

which tells us that his mother’s maiden name is WOOD and we find this marriage:
Marriages Sep 1904
Fluck Austin Henry Wheatenhurst 6a 727
Wood Adelaide Louise Wheatenhurst 6a 727

Austin FLUCK died:
Deaths Jun 1911
Fluck Austin H 32 Wheatenhurst 6a 207

and Adelaide married again:
Marriages Sep 1921
Fluck Adelaide L Townsend Wheatenhurst 6a 861
Townsend Frank Fluck Wheatenhurst 6a 861

Frank died:
Deaths Mar 1923
TOWNSEND Frank 54 Gloucester 6a 362

Probate granted to Adelaide Louise Townsend
Name: Frank Townsend
Death Date: 19 Mar 1923
Death Place: Gloucestershire, England
Probate Date: 31 Mar 1923
Registry: Gloucester

Mike

Hi Phil,
I'm pleased you actually knew far more than appeared to be the case. It was clearly a challenging search, and in retrospect I clearly should have waited for more clarification from you, sorry for leading the thread on a wild goose chase re Lily Gardiner.
Please, next time, can you tell us all you know at the very beginning.

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At least I found the correct William, thanks Mike for confirming my suspicions.

Following-on from the 1911 Census posted earlier, here's his father and grandparents in earlier years, all from the FamilySearch site.

1901 Census; Bristol Road, Hardwicke, Glos
Household Role Sex Age Birthplace
Henry Fluck Head M 54 Deal Porter Timber Yards. Hardwicke, Gloucestershire
Elizabeth Fluck Wife F 51 Westbury On Severn, Gloucestershire
Edgar James Fluck Son M 26 Groom (Domestic). Hardwicke, Gloucestershire
Austin Henry Fluck Son M 22 Deal Porter Timber Yards. Hardwicke, Gloucestershire
Sidney Charles Fluck Son M 15 Hardwicke, Gloucestershire
Sarah Edith Fluck Daughter F 14 Hardwicke, Gloucestershire
Winnie Gladys Fluck Grand Daughter F 6 Westbury On Severn, Gloucestershire

This record may possibly be investigated more fully using the FindMyPast site

Austin Henry Fluck
United Kingdom, Militia Service Records
Name Austin Henry Fluck
Event Type Military Service
Event Year Range 1806-1915
Event Place United Kingdom
Birthplace Gloucestershire
Birth Year (Estimated) 1879

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1891 Census; Green, Hardwicke, Wheatenhurst, Glos
Household Role Sex Age Birthplace
Henry Fluck Head M 44 Dock Laborer. Gloucestershire, England
Elizabeth Fluck Wife F 41 Gloucestershire, England
James E Fluck Son M 16 Under Groom Out Of Employment. Gloucestershire, England
Austin H Fluck Son M 12 Scholar. Gloucestershire, England
Sidney C Fluck Son M 5 Gloucestershire, England
Sarah E Fluck Daughter F 4 Gloucestershire, England

1881 Census; Hardwicke Green, Hardwicke, Wheatenhurst, Glos
Household Role Sex Age Birthplace
Henry Fluck Head M 34 Deal porter. Hardwicke, England
Elizabeth Fluck Wife F 31 Westbury On Severn, Gloucestershire, England
Martha C Fluck Daughter F 7 Hardwicke, England
James E Fluck Son M 6 Hardwicke, England
Austin H Fluck Son M 2 Hardwicke, England

I THINK this is possibly Henry's family

1871 Census; Hardwicke, Wheatenhurst, Glos
Household Role Sex Age Birthplace
James Fluck Head M 62 Ag labourer. Standish, Gloucestershire
Martha Fluck Wife F 64 Housekeeper. Quedgeley, Gloucestershire
Thomas Copner Stepson M 33 Ag labourer. Hardwicke, Gloucestershire
Mary Copner Stepdaughter F 27 Hardwick, Gloucestershire
Henry Fluck Son M 24 Ag lab. Hardwicke, Gloucestershire
Edwin Fluck Son M 21 Ag lab. Hardwicke, Gloucestershire

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Here's Adelaide's death record, confirming her birth c1877;

Surname First name(s) Mother/Spouse/Age District Vol Page
Deaths Sep 1958 (>99%)
TOWNSEND Adelaide L 81 Gloucester C. 7b 347

Adelaide Louise/Louisa Wood is not a rare name in London, but this seems the most likely fit for her birth record ?

Surname First name(s) Age District Vol Page
Births Jun 1875 (>99%)
Wood Ada Louise St. Saviour 1d 129

Ada is short for Adelaide, and the 1911 Census suggests her birthplace as Old Kent Road, London, which is in the district of St Saviour, Southwark.

Yet still struggling to find her in census' prior to the 1911, perhaps due to the limitations of the FamilySearch site and not seeing actual census images.
??