IMF villain Kurt Hendricks would be proud! (And this is a Frame Challenge.)
In your civilization's effort to rid the world of demonic blood, they destroyed the world. But I'd like to introduce another problem because, rather inconveniently, you didn't explain how you'd set of the nuke.
- Below the oil.
- Surface blast starting inside a large tank.
- Surface blast starting outside a large tank.
- Atmospheric blast.
Most, if not all, of the other answers are assuming atmospheric blasts. Let's introduce the idea of a surface blast.
Nuclear weapons convert a nearly meaningless amount of mass to energy. So we can statistically assume 100% of the oil is in play.
Heat? Oh, yeah, there's a lot of heat... but that heat doesn't exist in a vacuum. It's a nice thought that you could burn all the oil, but you'll end up burning almost none of it. A trillion barrels of oil is a lot of oil.
Barrels, not gallons or liters
The only oil that is guaranteed to burn is that oil inside the tank into which you place the nuke. (This all gets worse if you don't put the nuke inside a tank.) Because the first thing the heat will have to get through will be the walls of the barrels and tanks. Which aren't oil.
Which means most of the oil is being thrown away by the blast. I'm thinking Mount St. Helens. I'll get to that in a moment.
A trillion barrels of oil is 159x109 liters. A liter of oil will cover 7.5–10 square meters of surface. The surface of the Earth is approximately 5.1x1014 square meters. The oil will cover on average 13.9x1014 square meters.
Congratulations! You just covered the surface of the Earth more than twice over! WOO-HOO! Global genocide and destruction! However much is actually burned by the heat of the nuke is entirely irrelevant!
OK, I'm being a bit specious
Yes, that nuke will spread the oil far and wide. Really far and really wide. But even a 30 Mt nuke can't cast oil around the world. So, what would really happen?
Using the 1980's Mt. St. Helens eruption (approx 24 Mt) as a guide — and knowing that oil is a LOT heavier than ash — most of the oil will find itself within 50 miles of the blast center.
I expect people living +/- 1,500 miles away will experience some oil, but it's hard for me to imagine a significant amount of oil (e.g., enough to seriously affect crops) penetrating more than about 500 miles in any direction. Could be wrong about this, though. In any case, those nearby nations who opted to help your civ horde oil will never forgive them.
Mountains are your enemy (or everyone else's friends). Living on the other side of a mountain will be a definite plus.
The environmental clean up may never be fully accomplished. I can trivially believe breathtaking consequences for your world's oceans. Even a little oil falling over a vast area will have serious consequences.
What did burn, as mentioned in other answers, will create byproducts that will plague the world for decades.
Really, all your civ did was make the whole situation worse by proving once and for all that oil really is the blood of demons.
Is there a way to actually be rid of all that oil?
Oh, there may be bacteria or clever chemistry that could destroy the oil, but trillions of barrels makes both completely inapplicable. You have (IMO) one choice...
Incineration.
And you'd better have a plan for dealing with the combustion byproducts.
But I'll keep using nukes until all the oil is gone!
Congratulations! You just covered the surface of the Earth with radioactive fallout more than twice over! WOO-HOO! Global genocide and destruction! However much is actually burned by the heat of the nuke is entirely irrelevant! And the few people left alive, some of whom will be your your religious zealots, now look like this:

Photo of nuclear missile worshiping zealots from the movie "Beneath the Planet of the Apes"
OK, a surface blast won't work, what will?
Other answers have already pointed out the environmental consequences of an atmospheric blast. That won't spread the oil nearly as much as the surface blast would, but it would ignite the oil, which would be impossible to put out (they don't want to, anyway). But the toxic cloud would probably kill everything on Earth, anyway.
An underground blast would at best ignite the oil. It might not even do that. The deeper the blast, the less effectual it'll be. The closer to the surface you get, the more my surface blast scenario takes over.
Long story, short...
Cool idea unless you want scientific veracity, which I think is really boring. Nuke that oil! But if you insist on scientific veracity... burn, baby, burn. It's your only hope and unless you build a lot of hard-to-hide incinerators, it'll take a generation or two. Even an atmospheric blast would likely take years to burn away all the oil. If it gets it all (despite one or two or three more blasts).
Yes, science be a harsh task master.
I therefore bestow upon you the honorary certificate of the Rule of Cool. Go forth and nuke that oil!