I was delighted to see your post Robbo.
Since leaving my information I have filled in some of the WALKER family tree and believe that my husband's Lizzie (Eliza Ann) and your Lizzie are the same. I have not been able to find a birth in Ireland for my husband's great grandmother Lizzie, but did locate a suitable birth in Liverpool on 27th August 1884 and a baptism on 5th September 1884 at St Mathias, Liverpool. 2 other children were born in Liverpool - Mary (your line), who was not baptised with the name Josephine and Robert. I too have the other siblings you mention, all born in Ireland.
I found a marriage between Eliza EDDY and John WALKER and penciled this in on my tree and even found the widowed Eliza in 1901 and 1911 in Belfast. I had no idea where she was buried until seeing your post. My husband's grandmother was born in Belfast in 1918 but was subsequently given up for adoption back in London, so lost all contact with the family. Her mother was Eliza Ann(e) WALKER, who married Stephen Frederick Donald DAVIDSON in 1907.
Eliza Ann had 2 further children - Bernard and Josephine. I am sure that the younger sister's name shows another connection with your Mary Josephine. I have no idea why our Eileen was given up for adoption but subsequent children remained with the family.
In 1881 A John WALKER, ship keeper, is in Liverpool with 2 children (Andrew and Mary) and wife Johana. I think the wife's name is a mistake and it should read 'Eliza'.
I found the 3 young WALKERs in Liverpool in 1891, living with William and Elizabeth KENNEDY. I thought that Elizabeth was possibly Eliza WALKER (nee EDDY) who had subsequently remarried a William KENNEDY. It then looked as if the family returned to Belfast, where Eliza WALKER is recorded as a widow.
I hope this is of interest to you and you might be able to confirm some of the above.
Scurlockgirl