Just go to the Google Play store (assuming it's a Android phone) and search for 'NHS Covid-19'.
However, to the best of my knowledge, that app is only used to report the results of PCR test after you've experienced symptoms.
To report the result of a PCR test in an area when concentrated testing of everyone is being carried out, phone 119. (I've done that myself, as instructed by the staff at the test site).
To report the results of a later flow test, use the website that you seem to have already discovered. (I've done that loads of times. If you register your details, so that you don't have to submit them every time, it takes only a few moments after the first time. It's best if you use a device with a front-facing camera, such as a laptop or mobile phone, as you can just hold the test strip in front of the camera to register the number on it ).