Tesla obviously has the US market sewn up, BYD leading the pack in China, but unclear whether either of them will win in Europe. Euro-China JV's might be in with a chance, VW launching an made-in-china marqued for Europe EV soon.
Tesla’s about half the US market, BYD’s 35% of the Chinese one. If anything, you’d expect that Tesla’s current US market dominance is more a function of the relative immaturity of the US market than anything else; for various reasons it got mainstream-appropriate electric cars somewhat later than Europe and China, and the US market is the smallest of the big three, though it’s catching up. It would be reasonable to expect the US market to trend more like the European one over time; a bunch of companies with smaller shares of a larger market.