Other reports say a friend and fellow apprentice jockey was also on the scene. Ian Brennan jumped to safety when fire tore through the building in the early hours of the morning.
Brennan was later released from hospital and rode at Thirsk in the afternoon.
Another as yet unidentified jockey was also taken to York District Hospital to be treated for burns and smoke inhalation.
Kyne, 18, had ridden 29 winners this season and was third in the apprentice jockeys' championship.
His biggest career victory came at York in May when he won the sportingbet.com Sprint on Ishetoo.
Nineteen-year-old Wilson had been due to ride Imperial Sword, owned by her parents, at Thirsk on Saturday.
Tributes were made to the two young jockeys at Saturday's meetings where jockeys all wore black armbands.
Various horseracing organisations also expressed their sorrow at the news.
A British Horseracing Authority statement said: "Racing has lost two young jockeys who had already shown so much promise and talent. Our thoughts and sympathies are with their families and friends."
Kevin Darley, chief executive of the Professional Jockeys Association, said: "Jamie Kyne and Jan Wilson were well on their way in their careers as jockeys. The thoughts of everyone in the weighing room go out to their family and friends."