One gets only a passing glimpse of Elizabeth Williams. She is living alone with her daughter in 1841 census, and with same daughter plus grand-daughter in 1851.
1790. Born somewhere in Anglesey. The 1851 census gives her place of birth as Llanfihangel, but there are several parishes in Anglesey that begin with this prefix, and the addition of another word by the enumerator is impossible to read.
The top is the original scan, below is when I tried to enhance it
However if my take on who John Edwards, the father of Ellen Edwards actually was, then he was working at Llanfihangel Trer Beirdd in 1851 and 1861. Whether it is the same or, I do not know. It could be "Llanfihangel yn nhowyn" which is 7 miles west of Llangwyllog where her daughter was born.
1826 living in Llangwyllog, aged 37, when her daughter was born. Presumably that is near where John Edwards, then aged 38, was carrying out his trade as a nailor.
1841 census shows Ellenor Williams age 14, and Elizabeth Williams age 50 living alone in High St, Llangefni.
1851 census has a curious household of mother, daughter and granddaughter, all with different surnames. It can be seen later, when one follows Ellen Edwards life, that the granddaughter is indeed the daughter of Ellen Edwards.
Elizabeth Williams appears to have died, presumably still a a pauper, by the 1861 census. There are too many deaths of Elizabeth Williams for me to find the correct one.