We had similar. DC not coughing for a whole hour at a time, but coughing hard several times. It is "coughing a lot for more than an hour, or 3 or more coughing episodes in 24 hours", where the person has a new cough.
Our DC is certainly coughing and coughing, hard, more than 3 times in 24 hours. So DC has 1 of the 3 key symptoms, the others would be fever, (NHS sites define this variously as over 37.8,over 38, and "hot to touch") and loss of smell or taste. DC also has had other symptoms (but not key ones) associated with covid, such as headache, sore throat, short of breath.
So we ordered & have sent off a home test, which is justified by the cough.
To be honest, if we sent DC in to school, I suspect the teachers would be worried by the cough,even through a mask, and DC would be at risk of being sent home, which, being rather sensitive to implied criticism, DC would find unpleasant and embarrassing. Like this, there will be a result to refer to. If it does come back and doesn't get lost, that is.