There have been a couple of threads on this of late, but I can't seem to find them.
Nip/pop out are very British, they both mean the same, which is to go out for only a very short period. "Pop out" is more widely used, I think, whereas, "nip out" has a more regional flavour. But that may just be me.
"To go out" is quite neutral as to the length of the period spent out. It can be modified, of course, even adding "just" implies a short period: "I'm just going out for a coffee".