Hi
Civil Registration did not commence until 1855 so there won't be a birth certificate. I'm not sure why you don't think William is the father but the youngest children on the 1861 census, James born 1855 and Isabella 1858 were born after Civil Registration and are on Familysearch with parents named as William Candlish & Sarah Houston.
Familysearch for Scotland prior to 1855 are mainly baptisms and marriages for the Church of Scotland, the lack of entries for children of William would suggest they were not Church of Scotland. Most other churches are not online.
The 1841 census has William Candlish at what looks like the same address as 1851 & 1861. Also in the household are 3 Candlish children, John 11, Henrietta 6, and Catherine 3, and a servant Sarah Houston. I'm guessing that William was previously married and his wife died, he then subsequently married his servant Sarah Houston.
You really need to get the Robert Ellwood/Margaret McCandlish marriage certificate as this should give you her fathers name and occupation before you go much further.
Andy