Moses died on the seventh of Adar.
The Talmud proves this as follows:
In Deuteronomy 34:8 we read that the Jews mourned for thirty days following Moses' death in the Plains of Moab. This area borders Israel, just east of the Jordan River.
The book of Joshua begins with G‑d's command to bring the Jewish people across the Jordan River. G‑d specifies that they are to cross in three days time. This instruction was given immediately after Moses died, meaning at the earliest possible opportunity after his death. This would have been following the thirty days of mourning.
In Joshua 4:19 we are told that the Jews crossed the river on the tenth of Nissan. If we subtract the three days between the command and actual crossing, plus the thirty days of mourning, we find the date of Moses' passing is the seventh of Adar.
Incidentally, the seventh of Adar is also Moses' birthday. This we derive from what Moses said on the day of his death (Deuteronomy 31:2): "Today I am one hundred and twenty years old."
Malkie Janowski for Chabad.org
Would it be proper to mourn while traveling? I doubt it would be. So, if the host remained where they were for a full 30 days before traveling to the edge of R. Jordan, and if it had to take at least one day and probably more just to get there, we're already up to 31 or 32 days if not even more. Furthermore, evidently 'after' these days of mourning, and yet before moving the whole army, Joshua sent men to spy out the land and come back with their report. How long did that take? Between that and marching from Shittim to R. Jordan we're likely already up near 40 days give or take. And then even after all of that, where is it written that the army 'arrived' at R. Jordan on 7 Nisan? Receiving 'the command' on 7 Nisan is not one in the same as saying that's the day they arrived. The Talmudic description of things ignores some basic issues and makes a number of assumptions. The only objective conclusion that can be drawn is that Moses died 'no later than' late Shvat, possibly earlier.
We see from the first verse in Devarim, "These are the words which Moses spoke to all Israel on that side of the Jordan in the desert..." Meaning, they were already at the edge of the Jordan. Rashi explains that the other words in the verse are allusions to events that happened previously.
The Redak (Rabbi David Kimchi, 13th century) in his commentary to Yehoshua 3:2 mentions delays similar to your suggestions. Ultimately, he rejects them and adopts the Rabbinic tradition:
“…for it does not seem reasonable that the people of Israel would be waiting idly for a number of days after Moshe's death. Rather, immediately after the mourning was concluded, G-d commanded Yehoshua to traverse the Yarden…and on that very day Yehoshua told the people to prepare for three days' time…”
Yes, the plain of Moab is adjacent to R. Jordan, not even a days march. That's not really what's at issue here. Spies out while mourning is, which you did not address. But what's mainly at issue is the assumption that 'no delay' would be called for. Heavens, even God himself called for a 3 day delay lol. More to the point though, not rationally does one think to imagine that the complement of details provided is synonymous with all details available. Throughout history armies are known to set up camp for extended periods prior to making their move, and the fact that Scripture doesn't say they did so isn't logically to be taken as the equivalent of their not having done so. That the views of someone of religious prominence in deep history ran counter to that logic is not germane. Many such have said much absurd. A conclusion either has a logical basis or it does not. Perhaps you can cite his logic? It has to be more than "I see no reason why ..." I see no reason why there's gravity.