I do historical re-enactment as part of my job, so I have picked up a lot of accents through that. Basically, any accent I need/want to pick up, I find and listen to recordings of native speakers and I can usually imitate them pretty convincingly. Someone who knows what to listen for would be able to tell, but most people don’t. And I am pretty regularly mistaken for an actual Brit by actual Brits.
That said, I usually employ a pretty small subset of my repertoire for D&D. Mostly neutral American, RP or cockney English, Irish, and Scottish, plus various cartoony monster voices. Generic Skyrimesque “Norse” for dwarves. The usual.