When Nigella states "not buffalo mozzarella" please tell me what to use as the only other readily available one is a quite plastic ball! Could I substitute baby bocconcini? It sounds as though there is a "wet" mozzarella out there!
Fresh mozzarella is used uncooked in dishes such as salads and also cooked in dishes such as Nigella's Quick Calabrian Lasagne (from Nigellissima and on the Nigella website). The fresh mozzarellas are often sold in containers or bags with some whey, but if it is made locally and not transported far then it may just be sold in plastic wrapping. There is a much firmer, dryer type of mozzarella sold in blocks (usually vacuum packed in supermarkets) but this type is usually best grated and used for pizzas and Nigella doesn't usually suggest using this type.
Buffalo mozzarella is very soft and milky and is wonderful when fresh but doesn't necessarily cook very well as the soft, moist texture tends to make dishes soggy when it is cooked. Burrata is similar in texture to buffalo mozzarella and is also best uncooked. Nigella will tend to suggest regular fresh mozzarella and "not buffalo mozzarella" in cooked dishes as regular fresh mozzarella is firmer and easier to slice, and will let out less liquid as it cooks. Bocconcini are just smaller versions of the regular mozzarella so the texture will be very similar, and they are much more fiddly to slice up.