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What is it that makes peas get all covered in mildew?  ???

Is there a special treatment? 

Is this the same kind of thing that marrows can sometimes get?

Does on treat this condition?  Ignore it?  Will it spread if so? 

Or, do you rip out the plants/dig them in/compost the bits or burn them?

??? :-\ ???

I have it on both peas and courgettes... keep meaning to look it up

Mildew is a fungus and the spores for this travel through the air and, if conditions are correct, it will germinate on plants and cause an infestation. 3 tsp. of baking soda with a little dishwashing liquid used as spreader in 1gal. of water sprayed on the plants should get rid of it. Whether it is worth spraying your plants will depend upon their condition and how much more you expect to get from them.
There are many species of fungus that cause a whitening of plant leaves which are collectively called "powdery mildew". Whether the same one is infecting your peas and courgettes would need analysis by a laboratory.