Hugh Fearnley-Whittingstall is wrong - I realize this is a popular trope.
Specifically, cheese straws should be made with self-raising flour, not plain flour, and with roughly grated cheese (50-50 red leicester and strong cheddar works well) nor finely grated cheese. If you do it how he recommends, you end up with oblongs of dense, flat pastry. Do it my [sister's] way, and you get light, fluffy straws with nobbles of melted cheese throughout. Lovely.