Nickel is an important metal in itself for battery technology, but 2/3 of nickel goes into stainless steel, so from this perspective nickel isn't a very highly leveraged play on battery advances. Whilst Cobalt is a by-product of nickel mining, it is my understanding that this is mainly the case from lateritic nickel deposits, and there is a much lower % of cobalt by-product from deep mines.
Thank you for this educative email, from a subscriber who literally wrote the book on mining economics. I agree that not every nickel company produces cobalt which is why the three large miners I mentioned all specifically state they do produce it.
Among smaller companies Lundin and First Quantum both produce cobalt and have some of the largest reserves in the ground but other metals dominate their revenue streams.
Katanga is exploring for copper and cobalt in the Democratic Republic of Congo. The share has rallied impressively this year and continues to hold a progression of higher reaction lows. Despite the fact it is not a producer it is about as close to a pureplay as one is likely to find.
This article from mining.com may also be of interest.