We have a small potted Cox's Orange Pippin apple tree which is diseased but I have no idea what the problem is as the symptoms don't seem to match anything I have found on the internet so far. I have attached photographs and would be grateful for any clue as to what the pest/disease is and how to treat it.
I think your tree may have two ailments, not one. Cox's Orange Pippins are rather prone to diseases and deficiencies, I'm afraid. First, the spots on the leaves look like Cox's Spot to me. This is a physiological disorder and its cause is hard to pin down. It's usually attributed to drought and/or magnesium deficiency. Second, the immature fruit appear to have a bad case of apple scab. I'm not upto date on what's still legal to use but you need a fungicide that's approved for fruit trees. Your tree is still small so it'll be possible to give it a good hygiene regime. Cox's Spot and apple scab are quite common so if you've got a poorly Cox's Orange Pippin, it's likely to one (or both) of those.
Agree with the first diagnosis, but I don't think it is scab for the second. (Here is a link to a picture of at Apple Scab and instructions on what to do if it is). However, I think your apples are split, which is a sure sign of erratic watering or lack of water. Given that it is a Cox, I don't think a pot is the best place for it at all - they are famously tricky to grow and Flinty is quite right - they get every disease under the sun. I woudl try something like a Lord Lambourne if you like Cox's - much easier to grow. Good Luck