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I cannot find him in English censuses in 1901.  He was in England as he had just joined the police in Leeds and had previously been with some railway police for a year.  There is no sign of him anywhere in 1901, let alone as a police trainee or constable.

The principal clues about his birth are in the censuses:

In 1911 he said he was 35 and so born in 1876 and that he was born in Roxcommon Kilgefin, which has to be the Irish county of Roscommon.  And Kilgefin is the name of a civil parish in Roscommon (see <https://www.johngrenham.com/>;).
In 1939 he gave his birth date as 30 Mar 1876.  At least that agrees with the 1911 information.
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Can any Irish records be found for him?

The problem is which of the many Joseph McDermotts he might have been.  The birth place he gave in 1911 is recognisable for the Irish County of Roscommon, and Kilgefin is now recognised as a civil parish.

Doing a general search of the civil records on the Irish Genealogy site from 1864 to the present day, a search was made for births of Joseph McDermott from 1875 to 1876.  This produced six results:

Birth of JOSEPH MCDERMOTT in 1877
Group Registration ID 11719660
SR District/Reg Area Strabane
Sex N/R
Mother's Birth Surname N/R

Birth of JOSEPH MCDERMOTT in 1877
Group Registration ID 9465411
SR District/Reg Area Omagh
Sex N/R
Mother's Birth Surname N/R

Birth of JOSEPH MCDERMOTT in 1877
Group Registration ID 9949447
SR District/Reg Area Dublin South
Sex N/R
Mother's Birth Surname N/R

Birth of JOSEPH MCDERMOTT in 1876
Group Registration ID 9208695
SR District/Reg Area Westport
Sex N/R
Mother's Birth Surname N/R

Birth of JOSEPH MCDERMOTT in 1876
Group Registration ID 9208428
SR District/Reg Area Boyle
Sex N/R
Mother's Birth Surname N/R

Birth of JOSEPH MCDERMOTT in 1876
Group Registration ID 9208515
SR District/Reg Area Galway
Sex N/R
Mother's Birth Surname N/R

Birth of JOSEPH MCDERMOTT in 1876
Group Registration ID 9208592
SR District/Reg Area Dublin North
Sex N/R
Mother's Birth Surname N/R

Birth of JOSEPH MACDERMOT in 1876
Group Registration ID 9200422
SR District/Reg Area Dublin South
Sex N/R
Mother's Birth Surname N/R

Note that none of these were in Roscommon.  (Or were they?)

I then tried a search for Joseph Michael McDermott and none were found.

Searching for ‘McDermot’ made no difference.
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I then tried searches of the church records, baptisms only.  Nothing appeared for Roscommon again, probably because the records are as yet incomplete, though the coverage is still being extended.

I suspect the only way forward is to find where the Roscommon baptisms are held and go and search those.  But I can’t see myself doing this, currently.

The 1921 census may give a better spelling of the township where he was born.
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End of research into Joseph’s birth place and parentage.

Start of documentation for his family and after.
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In the 1911 census they had been married for five years and had had four children, all of whom were alive and were at 25 Aston Road, Bramley, Leeds, Yorkshire:

First name(s)  Last name  Relationship Marital status  Sex  Age  Birth year  Occupation  Birth place
Joseph Michael  McDermott  Head  M  M  35  1876  Police constable  Roxcommon Kilgefin Resident
Elsie           McDermott  Wife  M  F  23  1888  -                 -
John            McDermott  Son   -  M   4  1907  -                 -
Catherine       McDermott  Daur  -  F   2  1909  -                 -
Mary            McDermott  Daur  -  F   1  1910  -                 -
James           McDermott  Son   -  M   0  1911  -                 -

Note that Elsie’s age had been mistranscribed as 33 from 23 so I have corrected that and her year of birth.

1911 ref: RG14 PN26858 RG78 PN1543 RD499 SD1 ED2 SN168

Elsie’s surname can be deduced from the children’s birth indices:

Name:                      Mother's Maiden Surname:
MCDERMOTT, JOHN            SHARP  
GRO Reference: 1907  J Quarter in BRAMLEY  Volume 09B  Page 301
MCDERMOTT, CATHERINE       SHARP  
GRO Reference: 1908  S Quarter in BRAMLEY  Volume 09B  Page 298
MCDERMOTT, MARY            SHARP  
GRO Reference: 1909  D Quarter in BRAMLEY  Volume 09B  Page 267
MCDERMOTT, JAMES           SHARP  
GRO Reference: 1911  M Quarter in BRAMLEY  Volume 09B  Page 196

with the census names and dates being effectively the same as those in the registrations.

Later children were:

Name:                             Mother's Maiden Surname:
MCDERMOTT, ELIZABETH DOREEN       SHARP  
GRO Reference: 1914  S Quarter in BRAMLEY  Volume 09B  Page 418
MCDERMOTT, BERNARD MICHAEL        SHARP  
GRO Reference: 1919  J Quarter in LEEDS  Volume 09B  Page 495
MCDERMOTT, ROSALEEN PATRICIA      SHARP  
GRO Reference: 1923  J Quarter in LEEDS  Volume 09B  Page 734
MCDERMOTT, THOMAS DESMOND         SHARP  
GRO Reference: 1926  M Quarter in LEEDS  Volume 09B  Page 482
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In 1939 Joseph and Elsie only had the second, sixth and eighth of the above with them at 82 Seaforth Road, Leeds, Yorkshire (West Riding):

First name(s)  Last name(s)  DOB  Sex  Occupation  Marital status  Schedule  Schedule Sub Number
Michael J  McDermott          30 Mar 1876  M  Insurance Agent (Retired)  M  229  1
Elsie      McDermott          10 Feb 1888  F  Unpaid Domestic Duties     M  229  2
Bernard M  McDermott          20 Apr 1919  M  Clerk                      S  229  3
Thomas D   McDermott          28 Nov 1925  M  At School                  S  229  4
Catherine  Woods (Mcdermott)  02 Jul 1908  F  School Teacher             S  229  5

1939 Ref: RG101/3448H/016/15

Note that he called himself Michael J McDermott.  But his wife Elise and their daughter Catherine have the same birth details as on the 1911 census, so this record is definitely him.
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His record with Leeds city police survives and a copy of the page is available on Ancestry.

He started with the police on March 21 1900 - so he was NOT in Ireland at that time nor for the 1901 census..

He resigned on 26th July 1912.
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He was alive in 1939.  He then called himself Michael J McDermott.  His death cannot be found unless this is it:

First name(s)  Joseph
Last name      McDermott
Gender         Male
Birth year     1875
Age            67
Death quarter  1
Death year     1942
District       Bradford
County         Yorkshire
Volume         9B
Page           63
Country        England
Record set     England & Wales Deaths 1837-2007

The age of 67 would have been correct as he died in Quarter 1 and this was before his birthday on 30th march.

Would he have moved to Bradford?  Or did a relative report the death in Bradford?

It may be sensible to get the death certificate for this.  Ordered on 19 Aug 2021.
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The certificate arrived on 26 Aug 2021.  It clearly says that the age at death was 57, not the 67 on the index.  Further the death information came from his son George A McDermott of 3 Sunderland Road, Heaton, Bradford.  I don’t think for one moment that this record is for the Joseph Michael McDermott who come over from Ireland and became a railway constable.
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There is also a Michael J McDermott who died in 1952, but what was he doing in Surrey?:

First name(s)  Michael J
Last name      McDermott
Gender         Male
Birth year     1877
Age            75
Death quarter  1
Death year     1952
District       Surrey South Eastern
County         Surrey
Volume         5G
Page           819
Country        England
Record set     England & Wales Deaths 1837-2007

It is suggested on Ancestry that a death in 1952 of a Michael McDermott was his.

I shall order this certificate. Done on 26 Aug 2021.
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There is a Services record card for medal awards but on this it says “Pres Dead 22-12-14”.  This MJD i on the Commonwealth War Graves Commission rolls of war dead, with the service number of 3904 and death date of 22 Dec 1912 and commemorated on the LE TOUET MENORIAL, panels 27 and 28, France.  Definitely not him anyhow as he survived.
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Their marriage index:

First name(s)     Joseph Michael
Last name         McDermott
Marriage quarter  4
Marriage year     1906
MarriageFinder    Joseph Michael McDermott married one of these people
                  Elsie Sharp, Agnes Ethel Fisher
District          North Bierley
County            Yorkshire
Country           England
Volume            9B
Page number       436
Record set        England & Wales Marriages 1837-2005
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