I'm not sure if you're still waiting for information, but I was actually just looking at this book at my local library this past weekend. The relevant section begins on page 272, but unfortunately it does not include much detail about Daniel except stating that he was in the Prince of Wale's American Regiment and first settled in York County, New Brunswick. However, there is some interesting supplemental information on pages 212-215 describing the migration patterns of New Brunswick Loyalists to the Niagara region, specifically those of the Prince of Wale's American Regiment, which aligns with what is known about Daniel Cook's life.
Regarding Daniel Cook's wife, I have found a baptismal record for a Daniel, son of Daniel Cook and Catharine in 1775 in Dutchess County NY (which would match with your family lore). The sponsors/godparents are Daniel Cook and Mary Cook (so likely relations of some kind). Searches of the 1790 American Census show Mary Cook still living in the area but no sign of Daniel Cook and his wife Catherine lending credence to the theory that this record is for the baptism of Daniel Cook Jr, son of the Loyalist Daniel Cook Sr who would have already left for New Brunswick by this time. However, I have no idea yet what Catharine's maiden name may be (or if this even is definitely the correct set of Daniel Sr/Jr)...
I believe I am also descended from Daniel Cook/Lydia Churchill through their son Stephen, but I am having difficulty proving the connection with anything more than convincing circumstantial evidence. My line from Stephen Cook is through his daughter Lydia (1827-1907) and her husband Samuel Leetham (1818-1902).
"My" Stephen Cook was born in New Brunswick around 1805, lived his adult life in Ontario, and died sometime after 1880 (likely in Indiana or South Dakota). I know Stephen married a Lydia, but I am not sure of her maiden name... there is one index online that says it is Boldero, but I can't find any original source for that. Stephen's wife Lydia also listed her ethnicity as "Dutch" on a census record leading me to speculate she may be descended from Dutch settlers in the New York area who came to Ontario as Loyalists?
On the 1880 US census Stephen listed the birthplace of his father and mother as New York and New Brunswick respectively which certainly matches well with the Daniel Cook/Lydia Churchill couple. From the "First Families of New Brunswick" database it seems like this is the only possible Cook family in the area at the time with a son named Stephen, and most online trees I have seen match him up with them, but I would really like some actual documentation to support that.
I hope some small part of this will be helpful to you...