The Avnei Nezer reads the oath of the Shir HaShirim — "I have made you swear" — as addressed to all the lands of exile together, meaning the prohibition is against all of Israel rising up as a single body, not against any individual making aliyah: Avnei Nezer, Yoreh De'ah 454 states explicitly that individuals are permitted to go up, and the oath binds only collective, wall-like mass ascent.
Even so, Avnei Nezer, Yoreh De'ah 454 reasons that because the forbidden act — going up "like a wall" — would by definition consist of each individual going up, there can be no binding obligation on each individual to go up; imposing such an individual duty would effectively recreate the collective act that the oath prohibits, and those who proceed in defiance risk forfeiture of divine providence.
On the question of whether a grant of permission by the nations dissolves the oath, Avnei Nezer, Yoreh De'ah 456 firmly follows Rashi's ruling: "like a wall" means together by force (biyd chazakah), and if the nations grant permission the act is no longer by force — so ascent with permission is permitted, and he insists that no number of contrary homiletical positions (such as the Ahavat Yonatan) can dislodge Rashi's plain legal ruling on this point.
On the related question of whether there is any mitzvah to ascend at all in the present era, Avnei Nezer, Yoreh De'ah 457 records that R. Chaim held there is presently no mitzvah to go up, whereas the Rambam did not say there is no mitzvah — and the Avnei Nezer's own position is that a rabbinic mitzvah to ascend does exist, qualified only by the condition that one not go intending to act frivolously or to quarrel.
Mishneh Torah, Melachim and Wars 5:4 is not directly on the question of the oath but provides the broader halakhic backdrop regarding the seven nations, noting that their remembrance has already been lost — a factor the Avnei Nezer's analysis of the contemporary applicability of such oaths takes as context.