"The share of the total environmental impact of E-mobility caused by the battery (measured in Ecoindicator 99 points) is 15%. The impact caused by the extraction of lithium for the components of the Li-ion battery is less than 2.3% (Ecoindicator 99 points). The major contributor to the environmental burden caused by the battery is the supply of copper and aluminum for the production of the anode and the cathode, plus the required cables or the battery management system."
The copper and aluminium foils that make up the electrodes of the battery dominate. Both these metals are highly recyclable. Not much Li-ion recycling goes on currently as there's lots of different battery shapes + sizes, and there haven't been the economies of scale. I would imagine that whatever technology Tesla have standardised on, there will be (possibly closed loop) recycling plants.
But if the ease of installing these batteries makes home solar panels more attractive, that may offset their impact outside of EV use.