It looks like apple scab (Venturia inaequalis), which is a fungus. It's an airborne fungus that is spread by spores that land on the fallen leave, they survive the winter and then spread to the fruit.

Have a look at the leaves. Often with apple scab there will be lesions on the leaves as well as the fruits. It could be 'bitter pit' where the disease will go right through the fruit. This can be rectified with a treatment of calcium.

Thanks for both of your replies.
You are absolutely right about the leaves Jimmytheone; I had not noticed this. I have added a photo of one to my Question and as you can see it has brown specks on.
I cut some of the picked apples in half; the fruit flesh is alright. As for the ones still one the tree, I wonder if the damage will spread into them.
Do you think that they will be edible, and that they will taste alright?

I have done some research, and it seems that I will have to treat the whole tree with something like a fungicide, although I am loathe to spray my organic apples with anything. Do either of you know anything about this?

As you have cut some of the fruits in half and they are clean then Myron's first answer would have been correct. Yes, there is some signs of disease on the leaves, but they are not that bad. If as I have said had the fruits had disease running right through them, then I would have said 'bitter pit'. I would eat the fruit if you peel them, disposing the peelings in the dustbin. And yes, although you want to be organic, there is nothing worse than nurturing your trees for a whole season only to end up with some pest or disease; sometimes you have to bite the bullet and set in motion a spraying programme.