The William I have was, I think born in 1820. However I might be wrong. If it is the correct William he was a miner but that is all I know about him. I can check the information against a copy of a will that I have to see if I got it from there. It is definite however that Ann Mary was born in March 1822 and died in December 1900. It is also definite that John died in Worcester Infirmary in 1858 as I have a copy of his will and know that his sister referred to him (in a letter of August that year sent from the USA, when talking about his recent death) as "a miser" with "a belt of avarice around his waist that ruined him body and soul" (I have sought in vain to cast more light on this!) - he left nearly £117, a considerable sum! It enabled Ann Mary (married surname Rook(e)(a widow by then) to travel back across the Atlantic with her only living son to collect her share of the will before crossing the ocean once more to Poughkeepsie. For some reason I do not have his birthday recorded on my database but I can check the will to see if it states it there.
The brother George Adams was mentioned in the will. George was born on 10.11.1810 and died in the USA some time in the decade leading up to 1870. (He was still alive in 1860). The time span between the eldest and youngest is about 13 years the eldest being born in 1809 and the youngest in 1822 but I do not have the dates of birth for James and John who could have been older or younger.
Edward also emigrated to the USA, sailing on the same ship as George and his family who emigrated at the same time(1854). - Both these brothers went to the USA after Ann Mary Rook their sister.

P.S. they were a family of the RC faith, if this helps. There is an obituary written about one of their deaths in the USA which links the family to Lancashire but there is no indication anywhere else that this is the case, however, considering the religion there could be links at some stage with that county. In my family (Rook/Rooke) there is also evidence that someone in an earlier generation may have married an Adams and it is possible therefore that Ann Mary Adams might have been a distant relative of the man she married, e.g. a second cousin or something more distant. She writes to her brother and sister in law after her husband's death as "my dear cousins" which might though only be used as a term of endearment in the context in which it was written though it might also show that she is related through blood as well as through marriage.
The name Adams is also spelt Addams in relation to this family.