According to Rav Sadya Gaon (a native of Egypt) in his Tafsir to Numbers (34:5) this refers to Wadi al-Arish.
According to Rashi, (a native of France), in his commentary to Joshua (13:3), nahal Mitsrayim is the Nile.
הוא נילוס הוא נחל מצרים
Rashi is followed by Radak (a Spanish native of Provence) in his commentary to Joshua there.
This is also the assumption of Targum Pseudo-Jonathan to Genesis (15:18) who renders "nahal Mitsrayim" as "nilus d'Mitsrayim"; "the Nile of Egypt.
This Targum was likely written before Rashi; perhaps in the 8th century, making it the earliest of these sources, although it may date to centuries after Rashi (see here).
According to Radvaz, a native of Egypt, it is Wadi al-Arish, and the one who says it is the Nile is mistaken. (Hilkhot Terumot 1:7). See also his responsa (6:2206) in which he cites R. Eliyahu Mizrahi as being of the mistaken opinion that nahal Mitsrayim is the Nile.
R. Avraham Benedict in Kovets Beit Aharon V'Yisrael (1995 pg. 107) notes that it would be a little strange for the Torah to refer to major river like the Nile as נחל; a term that does not even refer exclusively to an area with water (it can even refer to a dry river bed).
Furthermore, Genesis (15:18) contrasts "N'hal Mitsrayim" with the "great nahal" the Euphrates. It is strange to call the Euphrates great instead of the Nile. The Nile is well over four thousand miles long, while the Euphrates is little more than seventeen hundred miles long. Furthermore, the Nile is on average almost three kilometers wide, and at the widest more than seven. The Euphrates on the other hand, is at its widest less than half a kilometer. Although the dimensions of these rivers may have changed, it seems strange to refer to the Euphrates as great, and not the Nile. Wadi al-Arish, on the other hand, is a frequently dry seasonal stream only about 150 miles long, that certainly doesn't match the Euphrates. (Cf. Rashi to Deut. 1:7 for a different explanation of the description of the Euphrates).
