WIlliam

William Oliver Colin Jones (Bridget was legally single at the time of his birth, thus his birth surname is almost certainly "Jones"): Is the son of Bridget Jones and Mark Darcy

As Bridget was legally single at the time of his birth, his surname was likely legally "Jones", though after her wedding there is nothing to say that his name wasn't then changed to "Jones-Darcy" or "Darcy".

Bridget Jones Baby
He is conceived following two different one night stands Bridget has with two different men. One being with her old lover Mark, and the other with a new man Jack Qwant. Bridget does not know who the father is until William is actually born, nor does she know the sex of the baby. Bridget, having just recently turned 40 when she found out she was pregnant was determined to keep the baby, despite the fact that at that time she was single.

He is born at the hospital, delivered by Dr Rawling, his mother's Obstetrician. At birth Bridget was single which likely gave William the surname "Jones". With William now born, a DNA test to determine paternity was able to be performed.

One year later, Bridget Jones finally happily marries Mark, but Jack (with no ill feelings) is still in William's life as a friend. While right after the wedding Mark clearly states, asking Jack "What are you doing with my son", it is never clearly stated if Mark is the biological father. As he is marrying Bridget, William would be his son anyways through marriage and adoption, so he could be referring to that, however due to the nature of the series it is likely that William is the biological son of Mark.

Trivia

 * As Bridget was legally single at the time of his birth, his surname was likely "Jones", though after her wedding there is nothing to say that his name wasn't then changed to "Jones-Darcy" or "Darcy".
 * William's mother, Bridget, had a pregnancy scare prior to getting pregnant with WIlliam, in Bridget Jones: Edge of Reason, again with Mark being the possible father
 * The night his mother went into labour with him, she got locked out of her house, the bank, losing her purse, having her credit card "eaten" by the ATM machine
 * Just prior to his birth, his mother was in relationship with Jack and broke it off just before he was born.
 * He played a part in his parents' wedding
 * For Colin Firth, actor for Mark Darcy, this is the second film (series) where he plays the role of possible father but paternity is never determined or clearly stated, the first was Mamma Mia (the first movie), where he played Harry, one of three possible fathers to Sophie.
 * It was revealed that this movie was supposed to be between Mark Darcy and Daniel Cleaver as it always has been through this franchise, meaning it was either Mark Darcy or Daniel Cleaver who could have fathered William. However, it was likely always going to be that Darcy was William's biological father. Hugh Grant refused to come back for the third movie so he was replaced with Patrick Dempsey playing the American Jack Qwant
 * In a unused scene in Bridget Jones: Edge of Reason, Bridget having a son was foreshadowed. In the scene she imagines giving birth to a fully dressed baby boy, with the tags still on the clothes, and being his mum all the way through to when he gets married, and at the alter he turns around to now aged Bridget and shows her he is wearing a shirt that says "I Love You Mum" under his tuxedo coat.
 * His mum arrived at the hospital at 8 cm dialated and was unable to get an epidural, however she was in labour for several hours
 * Dr. Rawling kicked both Mark Darcy and Jack Qwant out of the room when it appeared that neither was able to help Bridget through her labour pain
 * thus neither was present for his actual birth