Well, here's one idea. Let's forget about the whole electricity thing
and stick big wind turbine blade set directly on the washing machine drum!

But seriously folks... I can see this being an issue for lots of
electronic loads in any poorly regulated system. The intermittent
nature of wind, and the startup delays of diesel generators mean that
you need something in between to fill the (time) gaps. The classic,
but hardly cheap, approach would be a bank of batteries which are
inverted up to AC mains, with the generator(s) charging those.

A possible alternative is a bit of surgery directly within the
appliance. It would not be beyond the wit of (this) man to build a
small battery power supply into the controlling unit of a washing
machine. Its task is merely to keep the controller powered, not the
whole machine. However, this will be a different solution for each
machine, and will of course blow any guarantees out of the water.

Sadly, because this would add even a relatively small cost to every
machine, I doubt you'll find the major manufacturers interested in
tackling the problem, until such time as alternative and maybe less
reliable power sources become the norm.