My findings concur with CaroleW.
I think that her 'father George, Shoemaker' was either ficticious or indeed, her father was a George and a shoemaker but not actually a Young.
I think that she was probably born illegitimately to one of the daughters on the census posted by CaroleW.
Her birth entry seems to be:
Harriett Young Tadcaster Registration District (covered Bilton)
Septermber quarter 1848 ref. 23 658
Burgon (+ variants) has been even more difficult to search for.
Have found just one person to fit and age changes on each census.
The 1891 and 1901 Census confirm that he was born in Bradford.
1871 - Wilsden, Yorkshire (ref.RG10-4494-24-9)
John BERGEN widr. 54 Hand Wool Comber b. Queens Co., Ireland
William Bergen son 25 Masons Labr. b. Bradford
Alfred Bergen son 20 b. Worcestershire ??
1861 - Midgley, Yorkshire (ref.RG9-3295-118-38)
John Burgan widr. 44 Wool Comber b. Bradford (??)
William Burgan son 18 'can't read' b. Bradford
Joseph Burgan son 16 Worsted Spinner ditto
Alfred Burgan son 11 ditto
1851 - Bradford (ref. HO107 - 2308 - 139 - 31)
John BARGEN 32 Wool Comber b. Ireland
Margret " 31 ditto
Ellen " 13 b. Bradford
Eliza " 9 ditto
William " 7 ditto
Joseph " 5 ditto
Alfred " 7 mos. ditto
It would seem that William's birth was registered as Bergin Mar.qtr.1844 Bradford
(in 1851 shows as 1844, in 1861 as 1843, 1871 as 1846, 1881 can't find, 1891 as 1849, 1901 as 1852 - he died in 1905 with birth date as 1849)
Harriet Burgon died 1924 Bradford.
They married in Bradford - Dec.qtr.1883.
Annette