All utensils that are found going down to the mikvah are tamai and those found going up are tahor. R’ Yosee says that all utensils found in Yirushalayim are tahor except utensils used to bury people.

Shikalim 8:3

What we spoke about in the previous mishna, that utensils are tahor, is only for chulin.

A slaughtering knife, if found on the 14^th day of Nissan, is tahor and may be used to slaughter the korban Pesach. However, if it is found on the 13^th, we say you should put it in the mikvah to be used the next day, as maybe it needs to be immersed and is not tahor until that night. An ax, which cannot be used to cut the bones of the korban Pesach as it is prohibited even on the 14^th, must be brought to the mikvah as there was no reason for it to have been immersed.

If the 14^th came out on Shabbos and you can not bring the utensil to the mikvah on Shabbos, it is assumed it was probably immersed before Shabbos even though it is the 13^th.

If you find the knife on the 15^th it can be used right away. If the slaughtering knife is connected to the ax, we give the ax the laws of the knife.