I note that Charles Thomas Gigg was baptised on 13 Feb 1887 at St James Norlands, Kensington. His mother is given as Hannah Sarah Gigg, of 5 Mary's Place, single:
Source: Ancestry.co.uk, London, England, Church of England Births and Baptisms, 1813-1906
Whenever I have a case of a mother and illegitimate child vanishing without trace, my first thought is always that the mother probably married, and they probably appear on the censuses under another surname. That is what happened here.
Searching for a marriage on FreeBMD for Hannah Gigg between 1887 and 1891 brings up only a couple of options. Given her child's birthplace was Kensington, and the name given on the baptism (Hannah Sarah Gigg) this looks to be a likely bet:
However, we have a problem here. Only three names appear on this page, and there should be four. As luck would have it, Hannah's husband's name is the one that is missing. Nevertheless, it is a problem relatively easily solved. By performing a search of all marriages in Kensington RD in Dec quarter 1887, I was able to see that there was a lone entry on page 200: Thomas Walter Searle. Upon checking the original image, I was able to see that this was a transcription error, and the page should read page 209. Thomas was Hannah's husband.
More by luck than judgement I then came across Hannah and Thomas's marriage record in the London Parish Records on Ancestry.co.uk. It shows that they were married in St Helen's Church, Kensington, on 16 Oct 1887:
Source: Ancestry.co.uk, London, England, Church of England Marriages and Banns, 1754-1921
On Ancestry.co.uk, I came across a baptism record for Hannah Rosetta Searle, born 8 Aug 1888, baptised 2 Sep 1888 in St Mark's Church, Dalston, daughter of Thomas & Hannah Sarah Searle, of 102 Mayfield Road, bricklayer. Sadly this Hannah died in infancy. Her death was registered in June qtr 1889 in Edmonton RD.
I tried without success to find the family on the 1891 census.
However, they do appear on the 1901 census, at 13 Chalgrove Rd, Tottenham, Middlesex. Hannah is written as Anna, and there are four children noted: Annie (11), William (9), Lilian (7), and Emily (2):
Source: 1901 England Census, ref RG 13/1247, f 154, p 52
On the 1911 census, Thomas and Hannah and family are living at 42 Hartington Rd, Tottenham, Middlesex:
Source: 1911 England Census, ref RG 14/7328, no 199
Clearly Charles is not with them in either 1901 or 1911 – and he is married by 1911. I see according to his military records he married in Edmonton in 1908, which fits well with his mother and step-father living in Tottenham.
I suspect that wherever he was in 1891, he was with his mother and step-father, probably under the surname Searle. Whether it is a poorly transcribed entry or the family was omitted altogether, is difficult to say.
In 1901, since he is not with his parents, I would broaden your search to include various spellings of Searle. This seems to be a surname easily prone to mistranscription. In his youth Charles may have used Searle rather than Gigg, even though he married as Gigg.