His birth index, as Hogarth:

First name(s) JOSEPH HATCHELL
Last name HOGARTH
Birth year 1904
Birth quarter 3
Registration month -
Mother's last name -
District MARYLEBONE
County London
Country England
Volume 1A
Page 664
Record set England & Wales births 1837-2006
Category Birth, Marriage, Death & Parish Records

Ho! Ho!
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His apparent death, corresponding with the name given on his son’s Birth Certificate and his own marriage, entry in 1939 Register and Immigration record on entering US in 1946:

First name(s) JOSEPH HATCHELL H
Last name GAUTE
Gender Male
Birth day 21
Birth month 5
Birth year 1904
Age -
Death quarter 1
Death year 1991
District SURREY NORTH WESTERN
Register number 391
County Surrey
Volume 17
Page 986
Country England
Record set England & Wales deaths 1837-2007
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The next significant information is the Immigration record on his entering St Albans, Vermont in 1946 in the Ile de France from Southampton on March 30th 1946:

Family Name: Gaute
Forename:   Joseph
Age:   41
Sex:   M
Marriage?   M
Occupation:   Publishing
Launguages:   English: read and write
Nationality:   England
Race or People:  Scotch
Place of birth:  London
Visa Type:   Passport Visa
Issued:   London
Date:   Mar 12 1946
Last permanet residence: England, London

Noteworthy in the above is his ’Scotch Race’, the obvious next place to search.
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In 1911 he is to be found in the house of Herbert Everett-Gaute and his wife Lilla, being described as a step son of the former.  So Lilla was his mother and the mother too of his elder sister by two years, Iris Hogarth.

This led to his mother’s second marriage, giving her surname:

Surname   First name(s)      District   Vol  Page
Marriages Mar 1910   (>99%)
BRYANT    Lilla Elizabeth Westminster 1a 953
Burton    Lilla Elizabeth    Westminster 1a 953
GAUTE    Herbert Everett    Westminster 1a 953

So either Burton is her maiden name and Bryant is that of her previous marriage or vice versa.  But where does the Hogarth come from for her two children.  The plot thickens!
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This might be Iris’ marriage:

Surname   First name(s)   Spouse   District   Vol  Page
Marriages Sep 1928   (>99%)
Hogarth Iris Newton Kensington 1a 531

There are no Iris Gaute marriages in England and Wales between 1920 and 1960.
Newton Thomas H Hogarth Kensington 1a 531
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I can find no children for Herbert and Lilla Gaute, so her maiden name cannot be established by these means.
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There are no Hogarth to Bryant or Burton marriages in England and Wales between 1895, when she was 18, and 1905, when both her children had been born.
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Lilla’s death can be found:

First name(s) LILLA E
Last name EVERETT-GAUTE
Gender Female
Birth day -
Birth month -
Birth year 1874
Age 81
Death quarter 4
Death year 1955
District SURREY NORTH WESTERN
County Surrey
Volume 5G
Page 637
Country England
Record set England & Wales deaths 1837-2007
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There was a Herbert E Gaute death in Redruth, Cornwall:

Surname   First name(s)   Age   District   Vol  Page
Deaths Mar 1934   (>99%)
Gaute Herbert E 52 Redruth 5c 255

Short of getting the certificate, this is not certainly him, but could be.  It is the only Herbert Gaute death on FindMyPast’s World records from 1911 to 1991.
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Their marriage:

Surname   First name(s)   Spouse   District   Vol  Page
Marriages Sep 1937   (>99%)
Gaute Joseph H H Harbord Westminster 1a 1691
Harbord Alberta F G Hogarth Westminster 1a 1691
(1)  Additional information available - click on Info button
Harbord Alberta F G Gauts Westminster 1a 1691
(2)  Additional information available - click on Info button
Hogarth Joseph H Harbord Westminster 1a 1691

(1) On FreeBMD it says “Spouses name shown as Hogarth or Gauts”

(2) On FreeBMD it says “Spouses name shown as Hogarth or Gauts”

So this marriage certificate is also needed to see exactly what it says.

Apparently Joseph was even then using surnames of both Gaute and Hogarth.
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The 1939 Register shows:

First name(s) Last name(s) DOB Sex Occupation Marital status  Schedule  Schedule Sub Number

Joseph H H Gaute 21 May 1904 Male   ??? Married  108  1
Alberta F G Gaute 06 Aug 1912 Female Publishers Reader French Polisher  Married 108 2

A close examinationof the 1939 source document shows their occupations to be:

His: Publisher’s Editor: Foreign Films reader (French and German).  A.R.P. Warden (part time)
Hers: Unpaid domestic duties.  London Ambulance Service...
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