Well, lets try to get an estimate going.
Phobos has a mass of 1.065e16 kg, Deimos about 1.476e14 kg. Both of them are classified as Carbonaceous chondrite asteroids. Which is actually interesting on its own. According to Wikipedia, such asteroids contain high amounts of Water Ice, between 3-22%. So, lets say both have 15% Water by mass.
That would mean we get a total of 1.065e16kg * 0.15 + 1.476e14kg * 0.15 = 1.619e15kg of Water. Which equates to roughly 1,619,000,000,000 m³ of Water. Or 1.62 Trillion m³. Quiet a lot, but not really.
Earth has 1386 billion km³ of Water. All of the ice from the two moons would yield 1619 km³. Or 0.0001168%. Not a great start. And again this assumes both of them have unusually high amounts of Water in them. And that we can efficently process it AND make it not evaporate in the almost non exsistant atmosphere.
Now, a good question might be how much Water we actually need. This is actually a bit of a hard one to answer since we dont have a lot of data for this. But, if we want say 50% of the Surface covered in 10 Meters of Water on Average, we can get an idea.
Mars has a Surface area of 144,800,000 km². Half of that is 144800000/2 and we want all of this 10 meters under water so 144800000/2 * 0.01 (SI Units) = 724000 km³.
So, we can now see how much of that total our Asteroids provide. Which is 0.22%. Also not a great start.
And here it gets magical. If we are super good willing, we can say Mars itself has 1% of the Water we need trapped somewhere. In Ice, the ground or whatever. So, we need to find a way to get 98.8% of the rest from somewhere else. Thats a lot of Material which will have to impact the ground on Mars.
You mentioned using the Atmosphere for this. That is a horrible idea. We need the Atmosphere to prevent the Water from boiling off or worse becoming ice. Nobody is helped if we ship 3/4 of a Million km³ of ice to Mars only to have it freez into Pack ice. If you want oceans you need to first get the Atmosphere going to even allow liquid Water to exsist.
This is why Venus is a infinitly better candidate for Terraforming. Venus already has the Gravity to hold on to an atmosphere and a lot of Co2 which has uses. On Mars, you neither get Gravity or Co2. All you get is two pathetic moons and a toxic desert.
And i hope the numbers above illustrate the issue. Where do you plan to get basically 700000 km³ of Water from ? Sure, Ceres has more than that. But its all frozen in thick sheets and covered by millions of tons of Rock and Pack Ice. Thats the sort of Ice you need explosives to get going. The sheer energy envolved in Mining the ice would realistically halt the project dead in its tracks.
Another issue is heating. For all intend, you are planning to drop a 700000km³ Asteroid made of Water / ice on Mars. It dosnt matter if you do it piece by piece or all at once. That energy will have to go somewhere. That is a sphere of Ice 55 kilometers across moving at Interplanetary speeds. On Earth this would be an extinction level event not seen before. On Mars, which is a lot smaller than Earth, its even worse. You need to allow the Planet to cool down and deliver the ice with as little energy as possible. Which means long transfer times, very slow process and so on. Since you prosumably dont want to blow away the Atmosphere you are also creating.
All of this leads me to say that Terraforming Mars might be possible on a Time Scale of a few 1000 years. But if you try to do it in 100 it will fail.