Headteachers should recommend pupils and staff with covid symptoms to seek tests, according to the latest guidance.
They are also being instructed to avoid what officials refer to as the Scottish experience, in which the return to school prompted demand for tests from people with no symptoms.
A letter outlining headteachers' responsibilities has been sent to schools throughout the Swindon area and beyond.
It is signed by Yvonne Doyle, Medical Director and Director of Health Protection at Public Health England, and Susan Hopkins, Interim Chief Medical Adviser of NHS Test and Trace.
They say: "As schools and colleges return for the autumn term and you welcome back pupils and students, we understand that this may raise some concern about the risks of transmission of coronavirus.
"Along with all schools and colleges across the country, we know you are putting in every effort to follow the guidance and ensure that children and students can return safely as the new term starts.
"To aid this work we wanted to write with further advice with regards testing and shielding."
The letter says PHS has worked closely with NHS Test and Trace to minimise the potential risk of spread, and a key part of this is ensuring people can book tests online or by calling 119.
It adds: "Crucial to our overall support for schools and colleges is ensuring that everyone with coronavirus symptoms has access to a test as soon as they develop those symptoms.
"It is vital that we learn from the recent Scottish experience, where the return of schools saw a huge increase in demand for tests from people without coronavirus symptoms.
"We want to avoid a similar surge in test demand as England’s schools and colleges return to ensure that we continue to prioritise our testing capacity for those with symptoms and to avoid children, students and their families self-isolating unnecessarily."
Heads are cautioned to recommend testing for staff or pupils who develop one or more of the main symptoms - a high temperature, a new and continuous cough or loss or change to sense of smell or taste.
The letter adds: "If a pupil, student or staff member develops one or more of the main coronavirus symptoms described above, only they should be tested. There is no need for their households to have a test, unless they are also symptomatic.
"Unless a pupil, student or staff member has one of the three main coronavirus symptoms, they should only be tested if instructed otherwise by the local authority, health protection team or their GP.
"As schools and colleges across England return, pupils and students may feel unwell for example with a sore throat, stomach upset or a headache.
"These pupils and students don’t need to book a test but may need to stay off school or college and seek medical advice through their GP or pharmacist as usual.
"Please do communicate this to the parents of the children at your school and the students at your college; we know that they listen to you and trust your advice."
The officials also say even the small number of pupils and students who will remain on the shielded patient list can also return to school or college, and so can those who have family members who were shielding.