Heat the oven to 425°F and line a large rimmed sheet pan with parchment.
Toss the pulled pork with the barbecue sauce and microwave it until steaming, about 2 minutes.
Spread the chips in a single overlapping layer and scatter the pork and black beans evenly over them.
Cover with the cheddar and bake until the cheese is melted and the outer chips are lightly toasted, 7 to 9 minutes.
Whisk the sour cream with the lime juice until it can be drizzled.
Top the hot nachos with the tomato and jalapeños, then drizzle with the lime sour cream and serve immediately.