Connected my lathe to the oscilloscope with somewhat unexpected results.
Firstly, there is only a short spike at power on, VFD only, the motor is switched off. 
The spike is about 19A but it is very short. The current waveform immediately becomes a sine wave for several cycles over which the current gradually falls. Then short peaks, of about 0.6A, appear at the top of each cycle, although the waveform remains sinusoidal.

Switching the motor on makes the current peaks much larger. This is the motor running at 2500 rpm, with change gears, gearbox and drive shaft engaged, but the motor is otherwise unloaded.

This view show the current waveform to be dominated by sharp spikes, of about 9.6A each, and lasting about 2.4mS. As 50Hz UK mains takes 20mS to deliver a full cycle, the VFD only takes cuurent for about 10% of the available time. Presumably this is recharging the capacitors.
My VFD takes most current in sharp bursts at supply voltage peak. It does not spread demand for power across the supply waveform, nor does it soften the blow with any zero-crossing cleverness. The fuse is subjected to current spikes, not a gentle rise and fall. With the motor at 2500rpm the average current (measured with a wattmeter) is 3.1A but it is actually drawn from the mains in 9.6A pulses
Unfortunately I can't cut metal with the lathe and work the oscilloscope at the same time to see what a heavy load does to the current. Do the pulses get bigger or longer in duration or both? Don't know.
However, from the point of view of the 13A fuse:
- It has to survive a 19A pulse when the lathe is switched on.
- The fuse is subjected to a pulse load, not one smoothly proportional to mains voltage.
- When the lathe is consuming 3.1A on average it is actually taking power in 9.6A slices of about 2.4mS
- Unproven but, if the 3:1 peak to average is maintained, then the 6A average needed to drive a 1.5kW motor, is peaking at over 18A twice per cycle.
- Although the 13A fuse doesn't blow because it has time to cool, it must be stressed by this duty.
Going back to the original question, a 20A fast blow fuse looks like a better choice for 18A peaks than an ordinary 13A fuse. Although the 13A fuse might fail relatively early due to this mistreatment, it's not unsafe or unreliable. But pulse stressing an ordinary 13A fuse would be more problematical with a bigger motor, say 2.5kW: I think that's why the makers recommend high amperage fast blow fuses – a more appropriate balance of safety and reliability given all those sharp spikes.
My VFD is inexpensive. I suppose better electronics in a more sophisticated unit might load the mains less aggressively.
Dave