There's no lack of cobalt, it's only the most constrained supply. There's no problem with abundance as long as mines are opened.
Secondary problems of increased cobalt demand: the price of nickel will probably crash again, as they are found together except for in a few countries.
Seriously, though, like any commodity, dumping lots of excess nickel on the market could force mines to shut down if they can't weather the temporary explosion from a temporary demand spike in cobalt due to this lithium concept. Then when the explosion stopped, cobalt mining slowed down, you'd coast on excess reserves for a while.
When those reserves run out, suddenly you need nickel but the mines haven't been profitable for a decade, it takes time to restart surveying activity, procure specialized equipment, find/train employees...you can't turn it back on like a switch. But you can almost turn it off like a switch with new tech that obsoletes an industry.
Price crashes cause longer term price increases. The problem scenario is that a huge increase in cobalt demand could lead to a big oversupply of nickel as happened in 2007[1]. Mines are cash-poor so it only takes a few years of bad prices for most of them to go bankrupt and be forced to sell their assets. Since nickel and cobalt are mined side-by-side, the price of cobalt skyrockets to make up for the low nickel prices. Supply and demand bounce all over the place and the uncertainty makes it super unattractive to investors, which makes it harder to open or upgrade mines, which long-term raises prices.
Ideally careful pricing would avoid that situation, and most nickel mines will also be selling cobalt and will be well aware that nickel prices will be affected. However pricing is never perfect and the possibility of a crash is still there.
Also, lithium futures would get trashed, which would be bad for anyone invested in them.
Secondary problems of increased cobalt demand: the price of nickel will probably crash again, as they are found together except for in a few countries.