How can I remove all the gunk from the windscreen washer bottle.?

Nah you'd need something will a bit more kick. Give it a wash out with boiling water and caustic soda (available from any hardware shop). It wont damage the plastic, but will sting like hell if you get it in your eyes so be careful. Wear rubber gloves too.

It's not potent enough to damage the plastic but is harsh enough to irritate skin.

I use a lemon, one of those plastic lemons that are full of made up lemon juice. Empty into screen wash bottle and leave over night. It is a mild acid and will do no damage, in fact once you have it installed in your car, squeeze some into the bottle with water and it will keep the screen smear free.

They cost 29p instead of £3.99 for the same thing coloured blue and sold in garages.

Fill with hot water, squirt in washing up liquid and leave overnight.

I agree with t'other answer, plus use a bottle brush next day to scrub it out.

try a miled acid but leave over night as acid will not attack plastic