Having done the research if it's of no use to the original poster it may be of use to someone else.
I would point out that whilst we all work with given documented evidence it can sometimes be incorrect and in the below I believe this to be the case.
I will start at the end and work backwards with dates and links.
21st June 1918. The death of George Clayton of 423, Glasgow Road, Wishaw. Occupation Ironworker. Married to Agnes Williams nee' Gambles. Informant
Agnes Robertson--step daughter). George's parents recorded as David Clayton deceased & Elizabeth Clayton ms Rhodes
1911 scottish census
George Clayton 48 Ironworker born England
Agnes Clayton 43 born England
Joseph E H Williams 21 born England (Joseph Ernest Alphonso Williams mq 1890 Cockermouth mmn Gambles)
Arthur Clayton 23 born England
Agnes Williams 18 born England
Albert Clayton 6 born Cambusnethan (Born as George Albert Clayton 9.3.1905, father George Clayton, mother Agnes Clayton, Williams, Gambles. Parents allegedly married
20.10.1903 Workington)DQ 1902 Cockermouth Death of a David Clayton aged 53 = c1849
SQ 1902 Cockermouth Death of an Emily Clayton aged 43 = c1859
1901 England census for David, Emma & George in
Workington.https://www.familysearch.org/ark:/61903/1:1:XSV2-7K91901 census for Agnes Williams in
Workington
https://www.familysearch.org/ark:/61903/1:1:XSV2-4T11891 England census--
Newcastle under Lymehttps://www.familysearch.org/ark:/61903/1:1:WQYK-R3Z1887 Possible birth
POOLE, ARTHUR -
GRO Reference: 1887 M Quarter in
NEWCASTLE UNDER LYME Volume 06B Page 69 No mother's maiden name
1881 census for David Clayton--Note occupation-by himself
https://www.familysearch.org/ark:/61903/1:1:Q27V-SDHH1881 census for George Clayton--with an Emma She is born Shropshire as per 1891 & 1901 census so probably David's wife, it say "brother" but he is "brother in Law"
https://www.familysearch.org/ark:/61903/1:1:Q27V-Z5471871 census reference 4708 35 13 for George
1871 census reference 4708 47 37 for David & Emma
DQ 1869
Rotherham Marriage of
David Clayton to an Emma POOLE Emma aged 22 father Philip Poole occuption Waggoner, David aged 22 occ Shingler-father David Clayton occ Mill Furnessman. Married 25th December 1869 Parish church Rotherham. Image on Findmypast
1861 census, Rawmarsh,
Rotherhamhttps://www.familysearch.org/ark:/61903/1:1:M7G6-6QW1854 birth.
CLAYTON, GEORGE mother's maiden name RHODES
GRO Reference: 1854 J Quarter in ROTHERHAM Volume 09C Page 412
6th May 1849 Wellington, Shropshire Baptism of Emma Poole. Parents Philip & Rebecca abode Hayfield. Image on Findmypast
1846 birth
CLAYTON, DAVID mother's maiden name RHODES
GRO Reference: 1846 S Quarter in
WOLVERHAMPTON & SEISDON Volume 17
25th Sept 1843 David Clayton married Elizabeth Rhodes. Image on Findmypast
Please weigh up all this info, but after the death of David & Emily in 1902 Cockermouth I believe George Clayton took on looking after Arhtur Clayton(possibly Poole). I cannot answer why if David and Emily were married in 1869 why Arthur's birth is not recorded with mmn Poole but as George was always living with them could he have actually been the father?
John
Added--as per opening post
names being possibly albert- george-arthur-joseph
These names possibly being
George Albert Clayton 1905,
Arthur Clayton (Poole) 1887 and
Joseph Ernest Adolphus Williams