December 17, 1267:
Emperor Go-Uda of Japan is born. He died in 1324.
Take it away, Wikipedia:
Emperor Go-Uda (December 17, 1267 – July 16, 1324) was the 91st emperor of Japan, according to the traditional order of succession. His reign spanned the years from 1274 through 1287.This 13th century sovereign was named after the 9th century Emperor Uda and go- (?), translates literally as "later;" and thus, he is sometimes called the "Later Emperor Uda". The Japanese word "go" has also been translated to mean the "second one;" and in some older sources, this emperor may be identified as "Uda, the second," or as "Uda II."
His second son became Emperor Go-Daigo who may have been caught up in an adventure with the Doctor in 1336. As the Doctor explained to Lynda with a Y, "Then we went to... Kyoto.... That's right, Japan, in 1336.... And we only just escaped." ('Doctor Who' - "Bad Wolf")
More than likely, this happened during the Battle of Minatogawa, otherwise known as the Battle of Minato River. Those armies loyal to Go-Daigo fought against the Ashikaga Clan, who were eventually victorious. Go-Daigo fled to the mountains and the leader of his army committed suicide.
That's the most likely time in which the Doctor, Rose, and Jack met the son of Emperor Go-Uda.