Mabel, did Martha have any siblings? My Uncle believes that William and Martha were Ivy's Uncle and Aunt.
My best guess in that case is that maybe Ivy's father was Martha's brother?
Might explain Ivy having the Bateman surname as Martha had married William by then, and as you say, William had no siblings.
Maybe Ivy took William's surname as the man of the house, rather than her potential father's surname which would have been Lewis.
I can't see another explanation as to why Ivy ended up with William and Martha.
when Ivy was born there was no legal adoption process, so presumably they registered her birth as a Bateman with some untruths, if the suggestion that Alice MM Evans was actually her mother is correct.
My best bet is on Martha and Alice's mother Hannah being sisters or related - they are have the maiden name Lewis and both have in at least one census given their place of birth as Little Newcastle, Pembrokeshire. However it's not easy to follow them back with certainty in the censuses, unless you invest in some marriage certs to find out their father's name(s)
