Service records from that era rarely show any family details. If a man was already married when he joined up, that fact would be recorded, but this was fairly rare. The other place where dependants can be recorded is in pension records if a man is killed in action or invalided out; however there was no army pension for widows until the twentieth century.
Since Charles served exclusively in the British Isles (Ireland then under British administration) all births in England Wales and Scotland should have been registered. The starting points for official registration were: England and Wales 1837, Jersey 1842, Scotland 1855 and Ireland (North and South) 1864. Therefore there is a two year gap after their marriage until Irish registration began, but the
sources say that the battalion didn't go to Ireland until 1870 so this probably isn't relevant.
The complete list of deployments of the 2nd Battalion for the period is:
1857 England: Preston
1858 Gibraltar
1862 Ionian Islands: Corfu
1864 Jamaica
1867 Scotland
1868 England
1870 Ireland
1874 Guernsey
1875 England
1878 India
But as we know thanks to Shaun, he didn't serve overseas, so presumably he was at the depot during the years 1857 - 1867. Throughout that decade the regimental depot was at Colchester. This theory is perhaps borne out by the fact that when he re-engaged for a further 11 years service in July 1866 this was done at Colchester. If this is correct, you need to look at Kent births registered in Colchester.
An alternative explanation is that during most of that early period he may have been serving with the 1st Battalion. Their relevant deployment was:
Jan – Apr 1862 Bengal
Apr – May 1862 Passage home
June – Dec 1862 Gosport
Jan – May 1863 Gosport
Jun -Dec 1863 Aldershot
Jan - Sep 1864 Aldershot
Oct - Dec 1864 Plymouth (Devonport)
Jan – Jul 1865 Plymouth (Devonport)
Aug – Dec 1865 Jersey
Jan - Jun 1866 Jersey
Jul – Dec 1966 Cork
Jan – Nov 1867 Fermoy
Dec 1867 Passage to India
Incidentally, I don't think anyone has picked up on the fact that on page 2 (gbm_wo97_2013_051_002.jpg) of his record it shows he did 2 x 295 (590) days or roughly 19 months in the militia, presumably before he joined the regular army.