My question was rather different, much more simple/practical.
Today (not tomorrow, not ten years from now) I own a "normal" ICE car and for some reasons it breaks beyond any possible repair.
Currently (and it may depend on countries) I can sell the wreck to someone authorized to dismantle it and get a few hundred bucks for it or - in the worst case - pay someone to dismantle/recycle it a few hundred bucks (and I can - within limits - choose among many companies that provide this services according to the current Law).
What would happen (still today) if the car was an electric one?
The only difference is the battery cells might be stored instead of immediately recycled. (though, already there's a pretty hopping market for second hand battery packs from salvaged vehicles. Lots of people like to turn those into energy storage for homes/etc).
The simple practical answer is that storing batteries, right now, makes more sense than trying to immediately salvage them. Not because they are unsalvageable, but because future expected salvaging operations are expected to reuse more of the material.
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.
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
Batteries are resource gold mines with most material readily available (the casing, for example, would be the easiest thing to recycle).
Further, simple recycling (burn everything) of the inner parts is almost certainly worth it just to reclaim the current contents of nickel and cobalt.
What's not clear is if other resources can be reclaimed or if this can be taken further. Can we also reclaim the electrolyte (The substance between the anode and cathode)? Can we do all this without smelting down parts?
That's the unknown. That's also what manufactures are saying when they say "we don't know if this will be more or less valuable in the future." Because, as you can imagine, if recycling is simply applying a chemical wash to the electrolyte and reusing everything else, then you've saved $1000s of dollars avoiding making a new battery.
If, on the other hand, that's not possible, then smelting will still happen as that reclaims expensive materials (primarily nickel and cobalt) at a fraction of the cost of mining them fresh.
In other words, we know how to recycle battery materials. We don't know if we can recycle all of the battery materials. As a result, battery manufactures are just waiting until the cost of manufacturing new batteries becomes cost prohibitive without starting a recycling plant.