There are and were trade routes in history that went through the deserts, notably the Silk Road went through the Taklamakan Desert in Western China (Xinjiang province if you want to look it up on a map). Simply put, if there is a resource on one side of your desert province that people on the other side want, there will be a road through it or around it. Granted, I know nothing of your campaign but I would wager dollars to donuts that the trade route going through this desert region is at least partly the cause of the war. Since you outline two possible paths for the "peace" to take hold, we'll look at the scenarios individually.
The Hermit King
Likely if this war has lasted any amount of time, both regions have been using up resources on things that kill instead of things that make life better. If the hermit has any sort of power, he may be able to brow-beat the two factions into the peace, or maybe the warring countries pick him as the only person that each trusts to run things squarely. Regardless, he's now in charge and assuming he knows what he is doing, his first order of business would likely be to make sure that the region is as hospitable as possible. What I mean is he will want to figure out the feasibility of travelers/traders to make it through the area*. Are there reliable water sources about a half-day's travel apart? Oasis or town about a day's travel apart? If not, then figuring out how to get at least podunk towns at comfortable distances apart. Maybe in the dead center there is a 2-3 day slog through the desert, but have plenty of water and towns on each side so traders can stock up.
Trade will develop in 2 ways simultaneously. First, traders will expand into the desert by a trickle as they try to expand their trade networks. This trickle method starts with expansion to the "near" border of the desert, then a few brave souls will seek out new customers in the middle of the desert, and then moving to the far side of the desert, then to places beyond that desert. The other way trade will develop is that a noble or wealthy merchant will hear of a resource and finance a caravan to go load up and come back. They will likely have more gold to spend on things like food, bar maid's smiles and other "necessities" of life, but they are unlikely to want to sell much of their caravan's supply of the resource simply because Lord Fuzzy-britches would be most displeased if they brought him a sack of gold instead of the rare spices he ordered you to bring him back.
If both countries are annihilated, trade will likely take much longer to develop. I'm assuming that each country controls one side of the desert and the middle is something of a DMZ after this war. The first order of business would be that each community on either side consolidate resources and build/strengthen their respective towns/roads. As population expands, they will each likely send settlers (voluntary or not) to expand farther into the desert. When they meet in the middle, they will either work together or fight each other. If each country has a resource that the other country genuinely wants, this is the likely scenario for the two countries to form a somewhat lasting peace... although sooner or later the price of one of the resources will either go up or drop compared to the others "special resource". Likely there will either be hurt feelings or another war. However, that can also be centuries after your campaign ends if you prefer to not have a second war for that piece of desert muddy your campaign.
*Why would a hermit want trade/travel? Generally they wouldn't. However, once everyone trusts him to run things, the next likely scenario would be to try to bring in business to supply what the region cannot supply itself. As most campaigns are not long enough (read centuries), a little DM Fiat is a convenient plot-hammer to move the campaign in the direction you want.