Hi,
Williams birth is before the official registration for births in Ireland (1864). However Loudon is a fairly rare name in Londonderry and there was a couple, David Loudon and Anne Simpson, who married in 1857 in Woods Chapel Church of Ireland, parish of Artrea, Magherafelt district. Some of their children were Eliza Jane 1864, Mary 1865, Anne 1867 and Sarah 1873.
Regards