Sorry, but no, it is definitely not a Wilkinson bullet ...for several reasons.
1- The finder's photo shows it is only .75-inch long (or a tiny bit less). Wilkinson bullets were .95-to-.97-inch long.
2- The finder's photo shows it has a wide (and apparently bowl-shaped) cavity in its base. Wilkinson bullets had a simple flat solid base.
3- The seller says its circumference is "exactly 1 inch around give or take 1mm." Applying the Pi formula, that circumference measurement equals very near exactly .32-inch diameter. The only two sizes of Wilkinson bullets are .50-caliber and .54-caliber.
4- The finder says it has a reeded groove. (I can't see it in the blurry photos, but the finder specifically says it is reeded.) No Wilkinson bullets have a reeded groove.
5- Reeded grooves do not appear on bullets until around 1880. Wilkinson bullets are strictly 1861-65 era.
The "Handbook of Civil War Bullets & Cartridges" book says the made-in-America Wilkinson bullet was modeled on "a European design." So, because we can now be certain the finder's "reeded" .32-inch diameter mystery-bullet is definitely not an American-made Wilkinson, it would seem to be an 1880s-or-later European bullet.