It is a bank holiday so the rail service is either suspended or diverted because of engineering works. You are right – there is a regular national express coach between the two airports. You will find the National express office on the ground floor of South terminal by all the bus stops. It is just along from the public toilets. They will sell you a ticket and advise you of timings. You are then just at the mercy of the M25 although occasionally if the driver has alerted in time, he does come off and divert.

you can look at the National express website or download the app Best of luck!