Assuming you could 'light' the atmosphere on fire, it really depends on the speed at which it burns. Most combustion is subsonic - meaning the leading edge of the explosion is traveling less than the speed of sound. A good example of this is a trail of lighter fluid. If you light the end you can pretty much keep up with the leading edge of the flame by running.

If the atmosphere were a "high explosive" the leading edge of the flame/explosion would be supersonic. Since the atmosphere is about 20% oxygen and oxygen usually burns at subsonic speeds, we'll assume the atmosphere burns subsonic. The other assumption is that nitrogen in the atmosphere (about 78%) is inert and won't affect the burn.

If we ignite the atmosphere at point X, the flame will progress East and West at about the same speed (we'll ignore wind), up and down at about the same speed, and north and south at about the same speed. We're worried about the people of Earth, so up/down isn't really that big of a deal to us.

We've ignited the flame, it's traveling at 600mph, we'll say - because that's subsonic. The Earth has a circumference of about 25,000 miles at the equator, and about the same (within a few percent) around the poles.

At 600 mph it'll take 20 hours for the flames to meet at the other side.

We'd have a lot of issues trying this though. For one, the oceans are huge and as the fire burned across them they'd heat up and might release enough water vapor into the air to significantly slow or stop the atmospheric burning.

Another issue would be mountain ranges. The burn would fizzle out at some altitude where oxygen would become too thin to continue the burn. If a mountain range, like the Himalayan Mountains were in the way the fire may climb up the mountain but not make it through. Plateaus and other large land features might cause this same effect.

Furthermore, winds (which we ignored) would distort the flame's leading edge, especially on the large scale. At best this would slow the fire down, at worst it could stop the fire altogether.

There are also other issues that would prevent a combustion burn of the atmosphere. As the fire burned the air would heat up. The hotter air is less dense and goes up, which draws air into the bottom, feeding the fire from below. It's possible that you get a firestorm that doesn't actually move much. Instead, it just sits and burns intensely hot, sucking up air into its base and feeding the fire more.

If this happened we'd see a firestorm similar in structure to large thunderstorms/hurricanes. However, at some point it would burn out because its suction power is finite and it wouldn't be strong enough to suck air from the other side of the planet. In essence it would use up all the O2 from the surrounding area, then suffocate.

So the best guess, about 20 hours or so. The real guess, the atmosphere has been exposed to humans birthing small stars on the surface, to lightning, and to actual explosions and hasn't burned yet so it's pretty unlikely anything we humans could do would burn the atmosphere completely.