Hi,
Have you got William in the 1871 and the 1861 census?
Is William still alive in 1891 and 1901? If so have got you those censuses?
Do all these censuses just say Carmarthenshire for birth place?
what censuses do you have with William in? Is it just 1881?
One of the other censuses may throw some light on to where he was born in Carmartheshire
I took a look at the 1881 census , and there is no way I can jump back a huge gap of 30 years to 1851 looking for him with just that info you've given from 1881. All you seem to have is a name, no parents, no siblings names and an abt age for him.
Say for example if I find two or three 1 yr olds in Carmarthenshire in 1851- how will you know which one is your William, if any of them at all? ( he may not even be born in 1849/1850, and may even be born just after the census of 1851) . Searching through the whole of Carmartheshire, all the parishes, villages in 1851 would literally take months on end - it is not as easy as searching on a person's name like at the LDS's 1881 census, or at ancestry. To find people in most of the parishes in Carmarthenshire 1851 you need to look at each page - image to image, door to door. So you really do need to know what parish or village he is likely to be living in 1851- so the person can look through the pages at all the households at that particular parish and surrounds for you.
You need more info, parents or siblings names, and where he is likely to be living or born - you need to find him in the other censuses first if you haven't already . He should be with his parents in the 1861.
Have you looked for him in 1861?, That's easier to search, you can search by the person's name ( online). I don't really understand why you want him in 1851 first , if you have not got him in 1861 yet when he should be in that one too with his parents/ parent etc as an 11 yr old child. It's no good me looking for him in 1861. I wouldn't know if I've found the right one for you with no parents names to go on, no birth place etc
Again, you're going to have a problem knowing if you've found the right one in 1861 if you have no idea of his parents names , siblings etc, you'd just be guessing you have found him
You really could do with either William's birth or marriage certificate. A marriage should be the easier one to find first . To get Ann's maiden name why don't your order David John's birth certificate?If you can't find David John's birth, you could try looking for one of David's siblings births and order one of those instead.
The birth cert will give you Ann's previously known by name. with that info you can then try and search for Wiliam's and Ann's marriage . You could then order that marrige certificate and it will give you William's father's name and occupation ( who william says it is ) , at least then you will have some extra information to narrow down your search and may help to locate the right William as a child
Good luck, I do hope you find him. If I were you I would try and get some certificates to try and trace this family back through the censuses to get him in 1861 first as a child then back to 1851