If I am on the right track, Maria Mason (b 1892) married John B Mainwood in Walsall, Mar qtr 1919, and had a daughter Amy Mainwood in Sep qtr 1919.
Maria subsequently got together with Richard James Poole and they had a Poole/Mason son together in Wolverhampton in 1933. They belatedly married in 1949 (she as Maria Mainwood).
Maria died as Ann Maria Poole in 1954 with probate to her married daughter Amy Rawcliffe.
So was Bill adopted by the Pooles, or was he given up by Maria for adoption by a Johnson family? If the latter, his birth registration may have been as Mason or Mainwood.