If the Blight wasn't stopped when it was stopped, so what? Which warden would have the most at stake on a personal level to where they'd want to stop it? Particularly in the time period of the game?
Here's my take:
I honestly don't see the dwarven wardens caring much about Ferelden or even the surface world. If anything, I'm surprised that they wouldn't just bail and leave after Ostagar without anyone to stop them from deserting. Granted, Orzammar would be at risk and they have at least one family member still alive over there, but darkspawn encroachment has been a problem for dwarves with or without a blight. If anything, a Blight on the surface would be a welcome break for the dwarves, so why would a dwarf warden want to send the darkspawn back to the doorsteps of their people (assuming that they care about their people)?
Maybe a short-sighted moron would see it that way, but those with an ounce of forethought would realize that once the world falls, the darkspawn will turn back to them. The Blight is meant to wipe out all life on the planet, not just those in certain areas. Plus, darkspawn become much more cunning, organized, and rapidly expand during a Blight. In bluntest terms, the longer it goes on, the larger and stronger the hoard becomes. Why leave the horde to get stronger since the country they're expanding in is too busy fighting each other to pay any attention to it, risk it getting out of hand before putting it down?
In fact, the same goes for the Mages and Dalish. "Oh, I have no love for the Circle or the non-mages that put us there / my people have moved out of this country, so I can just get out of dodge and those I'm personally close to are safe." No they're not. The Blight goes after everyone, not just those in certain areas.
Not to mention that once the surface falls, all the trade, food, and goods that Orzammar relies on will go with it. Despite what the deyshers want to believe, Orzammar is not self-sufficient enough to go without the food, drink, and goods they have come to depend on. One little city-kingdom in a dark cave surrounded by darkspawn taint cannot sustain the population within. Even without the darkspawn showing up at Orzammar's doors in far greater numbers to flood them like a tsunami to an anthill, they'll be hurting pretty hard pretty fast without any surface trade practically overnight.
The City elf warden has a little more on the line with their home and family being in Denerem. Maybe they hate humans or not, but if Ferelden falls, then every elf in an alienage is going to fall right along with it.
I personally think City Elf Wardens have the most personally at stake. Denerim is their home, Ferelden their homeland, and city elves look upon the entire alienage as their family, not just their immediate nuclear relatives. Not to mention there are other elves out there besides just in Denerim. City elves have been largely socialized to look after each other ("We elves have to stick together"), so City Elf Wardens also have motivation to protect their people all over Denerim.
Human noble wardens seem to have the most at stake considering that the plot hits closer to home for them. Their country is the one hit with civil war, it's their country that's being invaded by darkspawn and it's his people who are being killed, eaten or worst.
How does the plot hinge closer to home for them than the others? They're not the only citizens of Ferelden; the city elf and mages are too. They're also not the only ones with countrymen and women threatened by the Blight. Not just other humans, but other city elves are threatened for CE Wardens, other Dalish elves for Dalish Wardens, other mages for Mage Wardens, and of course surface dwarves and Orzammar for Dwarven Wardens. In fact, when it's a human noble, how come everyone under the Ferelden sun regardless of race and magical ability is "their people" to feel concerned over, but if it's an elf, mage or dwarf, it must only be people who look and think and grew up right beside them?
In fact, in terms of having something personal at stake, I think the Human Noble has one of the least. Their entire family was wiped out, so it's not like they have many personal loved ones to look after. Their entire motivation in the game is killing the bastard that killed their family, which they do, then reclaiming the land, titles, and wealth their society says they're entitled to just for being born. Apart from getting the privileges and personal possessions back, what does a human noble personally have at stake that others don't?
I guess if they enter into an arranged marriage with Anora or Alistair, it becomes about protecting the country so they can rule it afterwards.