The wood used for scabbards comes from Villers-Coterets; hardly anything but beechwood is employed, it is bought in boards four pouces wide and two or three lines thick. After having been dressed with rasps it is cut with a knife along a steel rule in order to reduce it and also to divide it into strips suitable for the blade which is to be closed in it. These beechwood veneers are sold by the hundred.
No other mandril for making scabbards is used except the actual blade, upon which the wood is fitted as a preliminary operation, after which it is covered with linen, and finally with well pasted leather which is sewn on. After the whole assembly is well fastened together a metal end is put on the bottom and a hook on the top.Later, the process is described in greater detail:
They are made of beech wood, which comes to us in veneers from the environs of Villers-Coterets and a few other places, and are covered first of all with linen and afterwards with leather, shagreen, fish skin, shark skin, or some other material, in black, yellow, white, green, and other colors, well glued down.Article by Jacques-Raymond Lucotte, translated by J.D Aylward
The pouce was the French inch, about 6.6% longer than the English inch, the line 1/12 of that.
The 18th century technology of that article is a better starting for understanding medieval scabbard making than trying to deduce how they were made from first principles using our 21st century knowledge and intellects.
If you are trying to recreate the technology of, say, 1380, Diderot is closer chronologically than Sutton Hoo. And also closer than today.
Some excavated Anglo-Saxon scabbards used poplar or willow. The scabbard of so called Sword of St. Maurice in Vienna, probably from the last quarter of the 11th century, had an olive wood sheath.