Clutching at lots of straws here, but
assuming the birth was registered in the first quarter of 1858 in Cornwall (which it should have been but doesn't mean it was of course) then there is only 1 William James Thomas q/e March Redruth District vol 5c page 308. Looking at a list of places in Redruth Registration District and comparing that to the 1861 census (free from
http://www.freecen.org.uk ) with a William J/ William James aged 2 or 3 then there are these possibilities assuming the family hadn't moved of course
Tolcarne Street St Martin's CamborneJacob Thomas head age 27 or 29 copper miner Camborne
Sarah wife age 31 Devon Ashburdon ( Ashburton?)
Emma daughter age 12 work at tin mine Devon Ashburdon
William J son age 3 Camborne
Elizabeth A daughter age 1 Camborne
Jacob Thomas & Sarah Blewett married 21 May 1850 Camborne
Penponds CamborneHead of house Richard Allen, wife Ann children Robert & Lavinia
Elizabeth Thomas lodger age 25 Breage
William J Thomas lodger age 3 Camborne
Elizabeth Helen Thomas lodger age 1m Camborne
Fore Street RedruthJames Thomas head age 24 engine worker at a mine, Falmouth
Joanna wife age 25 Redruth
Mary Ann daughter age 4 scholar Redruth
William James age 2 scholar (?) Redruth
Margaret daughter age 3m Redruth
James Thomas and Joanna Thomas married 21 Feb 1854 St Just in Penwith
Ages can be notoriously wrong of course, but at least there is a starting point and if he was just shown as William in the 1861 census there are a few more I am afraid. If you ordered the birth certificate and put his known date of birth on the form then if that one in the first quarter of 1858 isn't the right one I don't think you are charged for it (not too sure but worth checking out)
You could also try - using freecen - following the William's above in 1871 and then if they appear in 1881 you can rule them out.
Good luck
Shirl