I found Elizabeth with her mother Annie in the 1939 Register. It gives Annie's date of birth as 1st October 1887 which might be useful and Elizabeth's marriage date as 4 September 1943. One anomaly I noticed is that the register has Elizabeth's date of birth as 23 December 1918 when other sources have it 1917. I've seen several off-by-one errors in the 1939 register before so this isn't a huge problem. I see an Elizabeth Davies with a mother's maiden name of Johnson born in Q1 1918 Bedwellty which fits with Dec 1917 and the register just being wrong.
Just to be sure that's the case, how do you know that Annie's maiden name was Johnson?
Looking for Charles Johnson (Annie's father) I came across Charles Underwood aka Charles Johnson, painter, whose master-criminal activities (stealing cheese for example) might explain why Annie and her sister were in the Nazareth House. I really like this candidate because I didn't include "painter" when searching for his name.