Might this be his birth index?

First name(s)  GEORGE WILLIAM A
Last name      AYNSLEY
Birth year     1874
Birth quarter  3
District       Bellingham
County         Northumberland
Country        England
Volume         10B
Page           317
Record set     England & Wales Births 1837-2006

But the 1891 census information makes this record unlikely to be his.  And Monkseaton, his birth location in the censuses, was never a parish within Bellingham district.
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No baptism record found, yet.
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Eventually he was found in the 1881 census as George Welsh, living with his grandmother at Fala Monkseaton, Monkseaton, Tynemouth, Northumberland (where was his mother?):

First name(s)  Last name  Relationship  Marital status  Sex  Age  Birth year  Occupation  Birth place
Ann     Welsh  Head       Widow   Female  69  1812  -        West Rainton, Durham, England
George  Welsh  Grand Son  Single  Male     7  1874  Scholar  Tynemouth, Northumberland

Ref:
Archive reference  RG11
Piece number       5083
Folio              75
Page               11
Combined:          RG11, piece 5083, folio 75, p. 11
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This 1881 census led to this birth registration index and delivery date of the certificate:

                Item                     Delivery   Despatch    Customer
Name            Type         Quantity    Method Date (Est)     Ref

GEORGE WELCH   E/W Birth     1          Upload      08 Jan 2020
                  PDF          

Year   Qtr    District    Vol     Page   Reg   Ent  No   DOR
1874   Mar    TYNEMOUTH   10B   182  

THe PDF of his registration was ordered and indeed showed that his father was unknown, his mother was Ann Welch, his birth date was 2 Jan 1874 and at Monkseaton which was his mother’s address and where she was the occupier of her house.  The birth was registered on the 2nd Feb 1874.
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The consequence of the 1881 census and the 1874 birth registration must be that George Welch’s father is not known.  His mother then married William Aynsley and George then took that as his surname, though at marriage both ensured that his birth surname was included and his father was given the name of his stepfather.
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Armed with his father's name from his, George's, marriage, this looks very much like him in the 1891 census at Whitley Park Cottage, East Parade, Whitley, Tynemouth, Northumberland:

First name(s)  Last name  Relationship  Marital status  Sex  Age  Birth year  Occupation  Birth place
William  Aynsley  Head           Married  Male    41  1850  Gardner  Matfen, Northumberland
Ann      Aynsley  Wife           Married  Female  49  1842  -        Monkseaton, Northumberland
George   Aynsley  Son            Single  Male     17  1874  Gardner  Monkseaton, Northumberland
Thomas   Martin   Nephew         Single  Male     18  1873  Butcher  Monkseaton, Northumberland
Mary     Welch    Niece          Single  Female   13  1878  -        Monkseaton, Northumberland
Ann      Welch    Mother In Law  Widow   Female   75  1816  -        Renton, Durham

Ref:
Archive reference:  RG12
Piece number        4230
Folio               104
Page                28
Combined:           RG12, piece 4230, folio 104, p. 28

From the above his mother's maiden name can be deduced from his father's mother-in-law's surname of Welch, not the Welsh that was given as his second name on both of his marriage registration index and his church Marriage record as transcribed by Family Search.  It is worth noting that no other records have been found where this second name of probably 'Welch' was used.

My guess is that the nephew and neice Thomas Martin and Mary Welch were something to do with the brother of George’s mother Ann.  In the 1871 census the brother Thomas and his wife Mary were living with Thomas’s mother Ann, the mother of Thomas and Ann the mother of George and also resident with George.  But Thomas and Mary have not yet been found, though they will be looked for, to complete the picture; they may have died of course.
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His death index:

First name(s)   GEORGE
Last name       AYNSLEY
Gender          Male
Birth year      1874
Age             40
Death quarter   4
Death year      1914
District        South Shields
County          Durham
Volume          10A
Page            993
Country         England
Record set      England & Wales Deaths 1837-2007
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No will or probate or administration was found for him.
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Their marriage index:

First name(s)     MARY
Last name         STOKOE
Marriage quarter  3
Marriage year     1897
MarriageFinder    MARY STOKOE married one of these people
                  Richard John Pattison, George Welsh Aynsley
District          Gateshead
County            Durham
Country           England
Volume            10A
Page number       1284
Record set        England & Wales Marriages 1837-2005
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And the transcription of ther church record by FamilySearch:

Name:                 George Welsh Aynsley
Birth Date:           1873
Age:                   24
Spouse's Name:        Mary Stokoe
Spouse's Birth Date:  1876
Spouse's Age:         21
Event Date:           08 Sep 1897
Event Place:          Venerable Bede, Gateshead, Durham
Father's Name:        William Aynsley
Spouse's Father's Name:   Thomas Stokoe
Marital Status:           Single
Spouse's Marital Status:  Single
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