Controlling 110v swamp cooler using Nest thermostat
Yes, it is easy to control a swamp cooler with any thermostat. The problem is that in order to use a swamp cooler effectively, you need strategic openings (typically open windows) to let the air out. This is the part which is hard to automate. The best way to use a swamp cooler is to open windows in the rooms that need the cooling the most. You want as much windowspace opened as possible without any outdoor air coming back in through the windows. There are some automated devices I've seen (such as UP-DUX that vent into the attic but they cause additional resistance so they are much less effective.

Your drawing has a couple of problems... 


You need to use the Rc terminal
The coil on your R8222B1062 needs to connect to the Y2 and the C (not the R)
I don't think you want to connect to the C terminal on the nest because your heating system is already providing the C to power the thermostat. 
I think you're right about this one but I wouldn't rely on the thermostat powering Y1 and Y2 simultaneously when it called for high. I would rather to design it so that if the Y1 is not powered but the Y2 was, it would still run on high. This gives you a little more redundancy in case of a relay or wire failure.


Below is a drawing that would fix these problems but it is much more generic, the only specific part is that Packard contact delay because I am unfamiliar with a generic part name for that. If you fix 1 - 3, your design should work flawlessly. Again, the exhaust ventilation is your biggest problem.



I would put a nice 24v DPDT relay and a nice 24v 3PDT relay on the appropriate mounting sockets and mount them to a DIN rail. This will make it easy and clean. Put it all in a box with the transformer and the Packard and voila.