For the full answer
I have already written a document that deals with this question in greater detail. The first four questions in the appendix from page 57 onwards address the Joshua side of things specifically, so I won't repeat myself massively here. I would recommend going to that document to work through the methodology for this and the whole Judges period (If you download the document you can view all of the graphics in glorious HQ).
I also have these two posts on here discussing the Judges time period:
When was the book of Ruth Set
Is There A Chronology Issue In The Book Of Judges
Here is the Briefest Summary
Judges is a math puzzle. 1 Kings 6:1 gives us a total of 480 years. Subtracting the years of Solomon's reign (4), David's reign (40), and Saul's reign (40), then subtracting the years in the Wildnerness (40) and the years of the conquest (6 as can be derived from Caleb's statements, they spied out the land in the second year). 480-130 years = 350 Years for the Judges period.
The total numerical data within (410 years adding oppressions, rests and judgeships) and around Judges (60 years in 1 Samuel 1-7) gives you 470 years of data for a 350-year period.
Judges 11:26 is the only event within Judges that is externally dated to another date; 300 years since the final year of wilderness wanderings, which would make 294 years since the end of the conquest. If we add up all data prior to this in Judges (18 years of Philistia/ Ammon, 22 Jair, 23 Tola, 3 Abimelech, 40 Gideon, 7 Midian, 40 Barak, 20 Canaan, 80 Ehud, 18 Moab, 40 Othniel, 8 Years Cushan) we get 319 years, which is 25 years too many, plus we need an indeterminate time between Joshua and the first conquest.
The linked documents point out, that we have case studies showing that judgeship data is overlapping data (as can be shown through Samson, Shamgar, and by reference Eli), only rests and oppressions should be used to build the chronology. When we recognise this, we find Jair and Tola are judgeship data (whereas Othniel, Ehud, Barak, and Gideon provide land "rest" data). Once we remove these from the equation, we find the solution for the whole period presents itself. We have 274 years of data instead of 319, which gives us a gap of 20 years from the end of the conquest until the first oppression.
With this method, we also find after Jephthah's 300 years, instead of 151 years of data to cram (which would encroach Solomon's reign by 15 years), we only need to fit 40 years of data in (the 40 oppression of Philistia), which gives a 16-year gap from the end of the Philistine oppression until Saul's reign.
Josephus gives 25 years total "reign" for Joshua (including the conquest), and 18 years from his death until the first oppression. Sedar Olam states 28 Years. Eusebius gives 27 years for Joshua. The Bible does not give a clear statement. Josephus gives two different numerical pieces of data to 1 Kings 6:1. He puts 592 years where 1 Kings 6:1 would be in his run-down/ summary (Book VIII, Chapter 3:1), but since he gives a different number (612 Years) when summarising this piece of data later in Book 20:10:1 (showing he disagrees with himself), it tells us that he is not quoting a number from his TaNaKh copy (a source of his history), but calculating this based on his own data. It seems in the second instance he includes the 20 years the Ark was in Kiriath-Jearim but omitted it in the first instance (if memory serves. You would have to find and add up all his data to double-check). If he was directly quoting he would give the same piece of data twice. Since he puts a number in the same place, it gives a strong indication that there was a number there in his copy of 1 Kings 6:1, but since he puts a different and self-disagreeing number in its place, it indicates to us that Josephus disagreed with 1 Kings 6:1, and so he just makes all the Judges data sequential with. He also omits Judge Tola entirely (and the enumeration of his total reflects this). Basically, his methodology is questionable and so I would question where he got any data regarding Joshua and the elders for this period of his history.
Presumed Same Age as Caleb?
Lots of people presume Joshua was the same age as Caleb, and so presume 25 years for Joshua, but this presumption is based on nothing. The Bible says that by the end of the war, Joshua was "old and advanced in years" and so was physically unable to continue fighting after only six years (Joshua 13:1). This same physical descriptor is given to him in Joshua 23:1,2. The only things Joshua needed to accomplish after six years of fighting were:
- Divide the land (Joshua 13:6,7)
- Build up his own inheritance in Timnath-Serah with his household (Joshua 19:50, 24:15, 29)
- Enter into God's "rest" as promised (Numbers 14:30, Psalm 95:11, Hebrews 3:11). It could hardly be said that he entered the "rest" if warfare continued and he enjoyed no peace before he died. So the nation stopped fighting until after he died (Judges 1:1).
These tasks would not require decades. Therefore, the time that can be deduced from the biblical data (20 years) is sufficient time for Joshua to grow old (chapters 22-24), die, Eleazar to die, Caleb and the other elders of the people (listed in Numbers 34:16-29) to die as the events of Judges 1:1-3:7, Joshua 13:13, 15:14-19, 63, 16:10, 17:11-13, 19:47 and Judges 17-21 to occur.
As to a specific birth year for Joshua, we can only speculate based on counting back 110 years from this 20-year window (which creates a 20-year window for his birth), and then you can count the generations from Joshua to Ephraim as contained in 1 Chronicles 7:20-27 to see if the math works (although this genealogy isn't 100% straightforward but that's another topic).