So just as an interesting aside (and not sure how accurate of an answer you're looking for), I chose to go with volume measurements just to see how far off from L.Dutch's answer we'd get. I believe that my answer is more accurate, but still ignores some things like the mass which doesn't need to be displaced by water above sea level as well as volumes below sea level not already filled with water.
So the volume we need to fill is the volume of the radius of a sphere at 6km higher than earth's surface (interestingly enough, right about the height of most rain clouds... go figure). From this, we subtract out the volume of the earth. This leaves us with a necessary volume of 3.77E9 km3.
Next we want to convert this to liters, so using that every 1km3 of water is 1E12 liters, we can see that we have a whopping 3.77E21 L of volume to fill.
Here is why I posted a second answer mainly... the metric system can be a pain for some things, but for the overwhelming majority of calculations, it is just the tops. 1mm in depth per sqm of water is exactly 1L of water. If the surface area of earth is 510.1e6 km2, this gives us 5.101E14 sqm.
Using the above, this gives us a rate (at 1mm per hour) of 5.101E14L/hr. Doing a little bit more math (dividing volume needed by fill rate), and changing to a rate of 30mm per hour to keep consistent, we actually wind up with ~28.1 years as the fill rate here.
The difference (which actually surprised me, I was guessing around 10% tops) comes from the fact that the total surface area (and more importantly, volume which is why it is drastically longer than the above answer) needing to be 'filled' will change as you get higher off of the earths surface. Filling in those gaps of square meter plates perpendicular to earth's surface will take an additional 6 years.