The wire is exactly the same. Amps are amps, regardless of voltage. So, whatever is in the wall works. If it's white, it's 14 gauge 15A; yellow = 12 gauge 20A.Ah yes, there you go. NEMA 6-15 or 6-20 ... same # of wires. Good stuff. I don't know if wires rated for 120V 15A are the same as those for 240V 15A though? If so, you can do what was suggested, change the outlet, leave the wiring, and change the breaker and the hookup at the breaker side and upgrade a 5-15 to a 6-15 and double your fun!
Tesla NEMA 6-15 adapter: Gen 2 NEMA Adapters
But, you must confirm the outlet is the only one on that circuit. Obviously, changing one outlet in a daisy chain to a 6-20 and leaving the rest 120V 5-20s would kill a bunch of appliances and start fires.
Here's the sort of breaker you'd use (for 20amp wire):
Square D by Schneider Electric HOM220GFIC Homeline 20 Amp Two-Pole GFCI Circuit Breaker, https://www.amazon.com/dp/B00L40PKFK/ref=cm_sw_r_cp_apa_i_-WiaDb2K0S0P2