I tried the American Breakfast Pancakes and I don't think these are fluffy American pancakes. They aren't thick at all. They are a little bit thicker than English pancakes. The batter is runny.
Nigella's American Breakfast Pancakes are made with a combination of baking powder and plain (all-purpose) flour. This is Nigella's version of an American-style breakfast pancake. In our experience in the US pancakes (also known as flapjacks, hotcakes or griddle cakes) can very in thickness and the thickness can be tweaked according to personal preference.
If you want to make thicker pancakes then it is possible to use less liquid to make a thicker batter. Try reducing the milk to 250ml (1 cup) then add extra a tablespoon at a time until you get the thickness you prefer. You can also let the batter stand for 5-10 minutes as this will give the baking powder more time to activate and you should start to see some bubbles forming in the batter as it stands. It may be helpful to check that your baking powder is still inside its "best before' date. Older baking powder will be less active and will lead to thinner pancakes.