What *fuses* an electric fuse?
Fuses blow due to heat, and heat is due to current via \$I^2R\$. Therefore, what blows a fuse is the current.

However, a fuse needs to blow with a large enough gap that the voltage behind current cannot overcome the gap. A fuse with a 1mm gap isn't going to be able to interrupt a current with 10,000V behind it (it might make a massive, long-lived arc that eventually extinguishes at best, and at worst the 10,000V will cause sparks across the gap or melt and weld the gap.) That's what determines the voltage rating of the fuse. The voltage rating is irrelevant while the fuse is not blown.

High voltage fuses do have different features such as sand or other materials that melts or buries the contacts when the fuse blows to reduce arcing or explosions.

As you found, fuse blow times are inexact and vary. If you look at the datasheets of some fast-blow and slow-blow fuses, you will find charts of how long it takes for the fuse to blow at a rated current. At 100% of the rated current, the fuse could take a very long time to blow (could run for minutes on end). At 200% or 500% of the rated current, it blows much faster. Don't expect a fuse to behave ideally and instantly blow within 1ms as soon as you exceed the rated current.

I cannot explain why it took 1-2 seconds to blow at 1600% overload though.