Fast blow fuses are used as line fuses in all Tektronix equipment, to cite an example. The initial turn-on current is not that great compared to the operating current, which should increase slowly on older tube-type equipment as the tubes warm up. On solid-state equipment, the initial turn-in current is a bit higher for a fraction of a second, then quickly settles down to a lower value. Don't get the idea that a fast-blow or instantaneous fuse is that. If you look a a fuse data sheet, you'll find that a 1a fuse will operate all it's life with a current of 1a through it. If a current of 1.1a flows continously, it may hold for several hours. If 1.2a flows continuously, it may blow after ten seconds. With 1.5a, it may blow after 1 second. The higher the current, the faster it will blow. So, it isn't truly instantaneous, blowing at the instant that the current crosses the fuse rating threshold.
Slow-blow (time delay) fuses work the same way, but on a much more extended scale. They may handle a 100% overload for 5 seconds; a 50% overload for 20 seconds; a 20% overload for a minute. These are hardly true figures for fuses, just typical representations of the idea that fuses don't blow quickly. For either a fast-blow or a slow-blow fuse, a 400% overload or a dead short will cause the fuse to blow instantaneously.