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I don't know about Lyme Disease.
Here in sub-tropical Australia, we have three species of ticks. They hatch on blades of grass just after rain, so the more rain, the more ticks.
Climate change melts ice and puts more free water into the systems across the planet. It increases the rate of evaporation and hence leads to more rain.
As systems become hotter they become less stable and less predictable. In sub-tropical zones, this means more cyclones of greater ferocity, but greater variations between seasons and years. But despite these variations the overall increase of rain is definitely producing more ticks in this part of the world.
Thank you for your helpful reply hartfire.
