Winchester did use other materials for stock making, but that said I’m unaware of there being any factory produced guns using English or Turkish Walnut.

They did use Birds Eye Maple on special order and Gumwood stocks during certain years of production. There also I believe were some guns were ordered without stocks and stocked by an outside entity. Therefore could have any sort of wood made stocks. 

The standard issue wood used was Northern American Black Walnut. In the Williamson book, I believe he discusses that in certain years when times were tough they used Southern American Black Walnut. Most people probably can’t tell the difference, but the grain structure is different due to the climate. 

The only thing close would be Winchester’s imported English shotguns, circa 1878–1884 to the best of my knowledge use English walnut. But these were already European produced firearms Imported by Winchester and merely stamped with the Winchester name. These double barreled shotguns were not big sellers and the practice didn’t last long for the company.

Sincerely,

Maverick