The prohibition on incisions in Israel is evidently postexilic.
In ancient practice gashing was a common sign of grief. Jeremiah 16:5ff gives quite a complete picture of the mourning rites. Although the prophet is here forbidden to follow the general custom, in order to show the divine cause of the exceptional impending disaster on Judah, notice no criticism is made about the practice itself, which is not set apart from other practices (such as burial, lament, or elsewhere ripping the garments) which were never criticized in later law:
For thus says Yhwh: Do not enter the house of mourning, or go to lament, or bemoan them; for I have taken away my peace from this people, says the LORD, my steadfast love and mercy. Both great and small shall die in this land; they shall not be buried, and no one shall lament for them; there shall be no gashing, no shaving of the head for them. No one shall break bread for the mourner, to offer comfort for the dead; nor shall anyone give them the cup of consolation to drink for their fathers or their mothers.
Aside from mourning, gashing was also a way of calling on the god's mercy (such as in 1 Kings 18:28): hence the association with priestly function (Leviticus 21:5). As such it was also accepted in Israel as part of Yhwh's worship:
They do not cry to me from the heart, but they wail upon their beds; they gash themselves for grain and wine (Hosea 7:14)
The Ishmael story in Jeremiah 41 is interesting from a number of standpoints, since it shows the "temple destruction" by Nebuchadnezzar in 587 had not stopped the sacrificial service. Notice the description:
On the day after the murder of Gedaliah, before anyone knew of it, eighty men arrived from Shechem and Shiloh and Samaria, with their beards shaved and their clothes torn, and their bodies gashed, bringing grain offerings and incense to present at the temple of Yhwh (v. 4f).
So if there is an Egyptian connection to the Torah prohibition, it is probably not older than the Exile (when many Jews fled to Egypt as the end of Jeremiah shows).
