The amount of work it takes to be able to dispel blizzards from your lands depends on whether you're neutral or not. Neutrals have it the easiest fighting against the Illians.
With access to Druids as neutrals, you have a unit that is capable of being produced and starts with revitalize (with proper +EXP amounts). It takes level 2 or 3, I forget, but I'm pretty sure level 2 druid (2 EXP) and you have a unit with revitalize.
For good or evil, you need to train mages and you need nature mana. This takes a lot more time. If you want to stay good or evil, make sure you've made a Great Priest generating city or two. Use every one of those Great Priests to construct Altar of Luonnotar. It's ironic that the civ with easy Altar of Luonnotar access is true neutral (Grigori) and they can never go good or evil.
Make sure that you have Ashen Veil Religion in the city you put the Altar of Luonnotar. Probably the best city to have Ashen Veil is your capital (and only your captain) to avoid losing this to Hyborem's whisper. When the AI uses it, it's kind of random, so usually you're ok, but it's best not to take chances. You then stack the crap out of +EXP (from civics or buildings) in this city. Build a Savant and you can almost upgrade it to an archmage (should be able to immediately with enough I think.
If you procured Nature mana, then you can now produce a unit which can immediately begin dispelling the effects of the deepening and whatever else crazy crap the Illians do. I don't enjoy using Scorch and Spring because they cannot fully undo the damage.