https://en.m.wikipedia.org/wiki/Antihydrogen

Anti hydrogen has been produced and detected in experiments. The energy transition levels are identical to normal hydrogen.

In a newtonian view, two particles orbit the center of mass of the combined system. Since protons are 1836 times as massive as electrons, the "orbit" center would be very close to the proton. So it's a bit like asking what would happen if we swapped the earth and sun. The orbits would change position, but the earth would still orbit the shared center inside the sun at the same orbital radius. So it would look essentially the same as it is currently, just with the center of the system having been shifted by one au.

Clearly I've ignored all of quantum mechanics in this description, but the conclusion is the same. The nucleus and electron both have wavefunctions, but the mass difference makes the spread of the nucleus negligibly small compared to electron orbits. Swapping initial positions and momentum doesn't really change the properties of the system.