No idea why you're posting this on the Boer War message board, and the Goodchild surname message board might be a better option
http://boards.ancestry.com/surnames.goodchild/mb.ashx but never mind.
Presumably you mean that he was a butcher by profession and not that Butcher was part of his surname.
IF he really was aged about 21 in 1951 then he was born circa 1930, and IF he really WAS born circa 1930 in England or Wales, then there are only 3 likely candidates plus some from nearby years.
None later than 1930 because the next birth for that name combination, after a 1930 birth, is for a birth in 1944, assuming of course that he wasn't also adopted.
However, just to cover the options, and ignoring any births before 1925, there are 8 candidates, and only one of them was born in Yorkshire.
Prior to that, the only other Yorkshire birth for that name combination was in 1903.
These are the candidates, the second surname is the mother's maiden surname, and the locations are Registration Districts, the years are the year of registration and aren't necessarily the birth years, because parents had 6 weeks in which to register a birth.
It doesn't matter though, because the following information is enough for the registration authorities to identify the correct birth registration record for each of them.
The single digit number after the surname is one of the four quarters of the year in which the birth was registered, e.g. 1 is the January, February, March quarter.
http://www.gro.gov.uk/gro/content/ARTHUR C. GOODCHILD 2 1925 PINCHBACK LIMEHOUSE London
ARTHUR J. P. GOODCHILD 2 1925 GIBSON DERBY Derbyshire
ARTHUR R. GOODCHILD 2 1925 HAY GUISBOROUGH Yorkshire
There was though a death in Doncaster in 1927 for that name and a very similar age which might be that child.
ARTHUR D. GOODCHILD 4 1926 HOOK KINGSTON Surrey
ARTHUR E. GOODCHILD 1 1927 CLUFF HACKNEY London
ARTHUR J. H. GOODCHILD 1 1929 STANNARD HARTISMERE Suffolk
ARTHUR W. GOODCHILD 3 1929 STOCKLEY PORTSMOUTH Hampshire
ARTHUR R. GOODCHILD 4 1930 BOWMAN WOKINGHAM Berkshire
However, even a birth certificate won't prove which of them was the father, and there's only one way that I can think of, that might link his address, to the address on your husband's birth certificate, it would also give his date of birth, and you'll find the information for that on the link below.
Those cards were in use up to February 1952, and whenever people moved, they were supposed to have their cards updated with their new address information, and they probably did, because the real intended purpose of those cards was as part of the administration of rationing, and rationing didn't fully end until 1954, and ironically perhaps, meat was the final food commodity to become non rationed.
Whether or not these records below were amended or not to include future address changes or not, I don't know, and of course, he might have been living at the same address in 1939 and 1951.
http://www.1911census.org.uk/1939.htmThere is perhaps a second way, if at the time, or in the following years, he was on the electoral register, and if he was living at the same address, and if there were any other family members living at that address, then that would be useful information, and electoral registers are listed in address order and not in name order.
The old Leeds electoral registers are available here...
http://discovery.nationalarchives.gov.uk/details/a?_ref=1564http://www.leeds.gov.uk/leisure/Pages/Local-and-family-histo...