such a long gap between james in 1907 and second child in 1920
Have you looked at FreeBMD? There are at least 6 Davies births with mother's maiden name Rooney in West Derby around the relevant time:
A girl in 1912
A boy in 1915
followed by Lilian (Lily) in 1920, John in 1921 and Sarah (Sally) in 1923 - these three you know about.
There's also another girl in 1924.
I can't post names of those we do not know are deceased but they are easily found on FreeBMD.
NB obviously it's possible there was another Davies/Rooney couple in the district responsible for some of these births.
As had been said, you need to look up the 1911 census to find earlier children. 1911 lookups (other than the free index) are not permitted on Rootschat.
Edit - sorry, I see Carmela has already referred you to the FreeBMD entries (at reply #4).