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Sorry this is so long-winded... but I'll try to layout what I know...
You are right on with my research, that Arthur Wilkins was likely born in Grimsby in around 1896:
His mother Lucy Wilkins b (Riley) 1862, in Rochford, Essex

Married James David Wilkins, b 1857 in Wilton, Wiltshire, England.
They were married and the shown in the 1891 census (before Arthur was born) to be living in Lambeth:

891 RG12 Piece:402 Folio:49 Page:54    
37, Conroy Street, Lambeth
WILKINS, James D Head Married M 33 1858 Painter Wilton Wiltshire
WILKINS, Lucy G? ('E') Wife Married F 28 1863 Laundress Rochford Essex
WILKINS, Ethel M Daughter F 6 1885 Aldershot Hampshire
WILKINS, David John Son M 4 1887 Pimlico London
WILKINS, Amy E S Daughter F 2 1889 Westminster London

sometime after this Lucy leaves Lambeth and moves to Grimsby (shown on 1891 census) but without James Wilkins.  Not clear what happened to him, whether he died, or Lucy just left I am not sure.  A letter from Lucy's daughter in her old age tries to explain that Herbert Barker may have been Arthur's father, but does not explain what happened to James Wilkins (which is part of the mystery), other than when she was little Lucy did not want to talk about him.

Here is what I found on Arthur G. Wilkins b 1896
WILKINS, Arthur
Caistor Lincolnshire 1895 Jul-Aug-Sep
Volume No: 7A Page No:  640

also:

SKELTON, Arthur George
Caistor  Lincolnshire 1896  Jul-Aug-Sep
Volume No: 7A Page No: 626

FreeBMD
Arthur George Skelton
Sep 1896 Caistor 7a 626

I have not found a marriage certificate between Lucy and George Skelton, but they do show up at the same address in the 1901 census with Arthur with last name of Skelton:
1901 RG13 Piece:3091 Folio:186 Page:36    
SKELTON, Geo Head Married M 38 1863 Fisherman Aldford Lincolnshir
SKELTON, Lucy Wife Married F 38 1863 Rochford Essex
WILKINS, Ethel M Step Daughter Single F 16 1885 Waitress Aldershot Hampshire SKELTON, Sarah A Daughter F 15 1886 Grimsby Lincolnshire 
WILKINS, David J Step Son M 14 1887 Nipper Lad Railway Westminster
WILKINS, Amy E S Step Daughter F 12 1889 Westminster London
SKELTON, Arthur G Son M 4 1897 Grimsby Lincolnshire
 
Unfortunately, I do not have copy of the actual original birth certificate which may prove who the father was, but all that I am going on is the letter years later written by Lucy's daughter "Amy Wilkins", that Lucy stated to her Herbert Barker was the father of Arthur G. Wilkins/Skelton.  How does one go about getting the original birth certificate for Arthur, with known mother Lucy?   Also, formal death certificate for Lucy's first husband "James David Wilkins" b 1857?

Thanks much.  Can you send me the transcription or copy of the census for Barker?  Wondering about his age, and place of birth.  Hoping to track down what happened to him.

Is this street directory available online? 

As you can see from the census records the actual date of Joseph  and Lucy is a bit of a mystery.  Since I have not located a record of birth or baptism for either I am guessing, but most other family trees that I have seen on Ancestry.com seem show Joseph Pennell b 1807 and Lucy Pennell b 1804 (Lucy's maiden name is unknown to me since I have not found a record of marriage).  I have seen other Pennells referenced on RootsChat.com but I have not been able to find any record linking Joseph to any of them.

Jane Pennell/Dadson, b 1829 (Jane PENNELL christened 4th April 1829 St. Botolphs Northfleet) appears to be their only child. At age 12 Jane is listed under the name as (Pennch) in the 1941 census living with a Ch Nichols, and Sarah Nichols.

Jane marries Henry Dadson b 1824 (married Dec 8 1850, St Botolphs Parish, Northfleet)

Jane had a child at age 17 before her marriage to Henry Dadson:
Elizabeth Pennell/Riley b Jun 28 1846, but she appears to have been raised by her grandparents (Joseph and Lucy)

1851 census
Kent, Northfleet, Dist1b, page9
Bow Street, Northfleet
Joseph PENNELL (Head) 41 b.1810 Chelsea Middlesex Labourer
Lucy PENNELL (Wife)47* b.1804 Cliffe Kent Char woman
Thomas ALDER (Scholar) 11 b.1840 Northfleet
Elizabeth PENNELL (G/child) 5 b.1846 Northfleet