Autumn Fruit

A couple of years ago we noticed some cherry-like fruit on the trees in the park. We didn't dare eat one, though they looked delicious. 
We took a walk through the park today and there are several trees bearing fruit, some nearly ripe, others still green. We opened one up and the stone is flattened, more like a plum. Does anyone know what they are?
(I've put them on the map, but if anyone picks them before me I shall be very cross).