I spent many games trying to get this one to work. The main problem I had was that research vastly outpaced the fungus' capability. I did find a method that worked quite well. I may be off on some of the names of the traits, but they should convey adequately which ones you need.
First, hopefully, you have the Darwinist and Ionized Helix. If you don't have these two unlocked, it just means that it will take you longer, but shouldn't affect the outcome.
Starting country doesn't seem to matter much. Just make sure its got ports and land borders to help with transmission. Your first mutations should be the Bird 1, Rat 1, Cold Resistance 1, Heat Resistance 1, Water Transmission 1 and Antifungals 1. These will help your fungus get started off. Make sure to devolve any symptoms that you get randomly mutated. This will give you more DNA to get your fungus out there.
Eventually, you'll have enough points to get Bird 2 and Rat 2 selected. Occasionally, you'll need to use a Spore Burst to get things going in other places. However, there should be a decent transmission just from the ships and birds. Keep devolving symptoms.
If you make sure that your fungus is totally benign, no research work will be done for the cure. Keep time moving forward, popping bubbles as they come up, and storing up DNA points.
Once the planet is completely infected, you can devolve your transmission traits along with the heat & cold resistance along with antifungals. You don't need them since everyone is infected. Make sure you have enough DNA points to get a large number of symptoms. Start buying whichever tree you want, just make sure to get the higher level symptoms.
Your mortality rate should be extremely high at this point. It will outpace any research that can be done. Good luck!