One used to be able to trace adoptions through the Family Records Centre which used to be situated at Farringdon, London. That building has closed down and now everything is at the National Archives at Kew and I imagine all the records have been transferred there including the adoption records, which whilst doing family tree searches, I was able to examine them and I found what I wanted. Before trekking down there, it might be worth ringing them on 02088763444.
One of the problems with searches at Birmingham is that they seem to direct you on their computers to Ancestry. The first time I visited, the Library was the old concrete one on Chamberlain Place and I was shown microfiche, and record card records as well as books that had a great deal of detail with regard to what I was looking for, the last time I came up went to the new library and all that seemed to be availablke were computers linked to Ancestry to which I already subscribe. For that reason I have never been able to find a marriage, assuming there was one as they did not seem to have any church/parish records available.
Bob