Joseph was still around in 1901 - see census details previous page - in Allerton Bywater. 1906 he is in his mother's Will and is referred to as Joseph (no W). 1871, 1891, and 1901 he doesn't use the W so we don't know if this was a red herring when he married Margaret Lynn, or did he really have a father Joseph W(Wingrove, Wingrave) Smith. If so we never found him!! Going by where his children were born - and he seems to keep changing his mind about that as well as how old he is - the family moved around.
To ascertain if he is one of the three you give would mean buying certificates I think, to see who registered the death - but that's no guarantee that it was one of the family.
I haven't has yet looked for him aged between 53 and 63, in 1911.
I rather think the W part was added in to make him "legitimate". His wife was very young when they married - only 16 I think.
Margaret