Well, my windscreen washer stopped squirting again. Headlight washer was fine. So, bit the bullet and took out the bottle and pumps etc. with the assistance of the DIY info above.
Windscreen washer pump filter was covered in brownish goo but the headlight washer pump filter was pretty clean - probably because it is a few centimeters higher in the bottle and all the gunk settles to the bottom around the other pump.
One point on the top DIY video - the large "rivet" under the sill is actually a plastic screw sort of thing and is a torx 27. So unscrew it rather than pulling it off. I also managed to find the three expanding rivet pins so could reuse them.
I put it all back together, filled with demineralised water and now both washers blast the water out.
Maybe it'll be good for another 8 years....