The 1901 census certainly looks correct with the birth places transposed between Ann and her grandson Frank.
The 1891 census would be right showing a Ralph Lucas as a grandson as Ann's second eldest daughter Mary Ann was married to Henry Lucas, but poor old Edward now says he is born in Birkenhead. If he has stated rightly about Birkenhead on this census then the other info including his two son's and wife Mary that you have found for the 1861 census looks very positive and he was actually a nearly local lad who had ended up in Liverpool but not from Manchester as originally stated. On a more romantic note as Ann already had a daughter named Mary Ann would she have called their daughter Mary Ellen or was this for Edwards first wife, she seems generous enough to do that.
The 1851 census showing Edward with father as John a tailor with birth age tallying looks great as well. All together looks as though you may have cracked a mystery.
I wonder why Edward gave different areas as birth places on census or are they all the mistakes of enumerators with the biggest one of Edward's two son's being born in the West Indies. This whole search started off with Ann and her first husband John Hargreaves who I started to wonder about when I realised she was a widow at a young age on her own in Liverpool with six children the youngest shown as months old being born in Manchester, I believe I have found John's death in Liverpool. And then I found Edward, it was great that Ann and Edward met and seemingly had a long marriage and helped each other over what must have been a lot of sadness for both of them but from their marriage we now have another name of Cunningham to add to the list.
Thanks once more to all, for all your hard work, far more than I expected cheers L.B
gave me a right headache 
turned out she was born in Wrexham, but her mother, a Rowland was born Manchester 
... yes good that he found Ann in the end 