Who (names of the companies) today (not tomorrow, not in ten years time) will pick up my wrecked electric car (and its batteries)?
For an old, traditional, ICE car I open the local phone directory and can find a handful of car wreckers that are authorized to scrap the car.
What do I do if the car is an electric one?
So, pull up the phone directory and call any one of those wreckers, ask them if they handle EVs. All of them will say yes.
Here (Italy, but I believe that several EU countries are in this same situation) they won't touch an electric vehicle with a ten foot pole.
They are not authorized to handle or store lithium batteries (that are considered dangerous wastes) and don't have the (safety) qualifications nor tools to handle mid-range voltages.
There are in the making some new EU norms (and investments) related to "circular economy" for the EV batteries:
https://ec.europa.eu/commission/presscorner/detail/en/IP_21_...
but right now they are e-waste, the so-called RAEE/WEEE, and while there is a code 160005 (that is for batteries coming from phones, notebooks, etc.) EV batteries are likely to fall in the 160215 code (which is dangerous parts removed from assemblies).
We are talking (if you can find anyone that will dismantle them) of roughly 4/5 Euro/kg for the disposal fees.
There's a pretty big second hand market for salvaged batteries. Except in the case of catastrophic failure, these batteries are either being stored for later recycling, or being sold outright for a second use unchanged.
They are not ending up in the junk yard. [1]
[1] http://store.evtv.me/proddetail.php?prod=TeslaBattModule
Tomorrow(ish): We recycle more of it.
But your question in many ways is less practical. Statistically cars have lifespans that we can chart out rather well. We can see that we have X tons of old batteries today, and on some tomorrow we'll have X+Y tons of old batteries. Which is important. Industry tends to scale. Working with 100 tons of material may not be an economical in a batch process. But when you have a million tons of the same materials you build an entire facility around said process and the small efficiencies add up at every step.