I have an update on this. Some new records came on line and I found Banns for a marriage of Josiah BRIGHT and an Elizabeth DOUGHTY in 1828/9 at St Mary, Whitechapel (London). Josiah BRIGHT does not seem to have gone through with the marriage as I can't find one for this couple in parish records. As was normal for this period, no ages or addresses are given. In 1829 a George Edwin BRIGHT is baptised at St Luke, Finsbury, London, a few miles away, with the father given as Josiah BRIGHT, a miller, and Mary Ann as the mother. In 1841 George DOUGHTY, age 11, is in Greenwich with Elizabeth DOUGHTY, who says she is his grandmother. When George DOUGHTY is married the certificate shows his father as Josiah BRIGHT, a miller. I have found a Josiah BRIGHT, who came from a family of millers in Buckinghamshire. he was single at the time of George's birth, but there is nothing to suggest he came to London. He later married and ended up running a pub. George DOUGHTY became a brushmaker, and I have found an Elizabeth DOUGHTY, the daughter of William DOUGHTY, a brushmaker of Whitechapel which could explain this. William had another daughter, Mary.
All this happened before proper census records or Registration began in England so you can only guess at the situation, which is possibly this: Elizabeth DOUGHTY, daughter of a Whitechapel brushmaker, meets Josiah Bright, who is a miller. She falls pregnant by him and a marriage is set. Josiah absconds or dies and Elizabeth has to raise the child alone. It was unheard of not to baptize a child. It was believed an unchristened child's soul would go straight to Hell if it died! She takes the child to Finsbury to have him baptized and gives the real father's name but a false (or possibly her sister's) name as the mother. It may even have been her sister who did this. According to the 1841 census she was about 30 when George was born so this gives her the ability to pose as his grandmother, instead of his mother. It may even have been more complicated than this! As for Josiah BRIGHT, the period prior to 1837 when registration began, and 1841, the date of the first proper English census is very, very difficult to research, unless the people concerned were rich. Ages, addresses, maiden names and lots of other information relating to ordinary people is not given on parish records. This means that proving a particular name on a page is the person you are researching is very difficult. Unless something relating to Josiah turns up, this will probably be unsolved.