The Sod Moth! Crikey, there are a few of those around - just ask Tea Drinker.
Sorry, to be serious, your London lawn has just endured a very hot spell and is taking its time to think about getting going again.(Unlike our lawn which has been well rained upon for most of your London drought!). Your lawn will recover when the rains begin again. If it makes you feel like you are doing something useful, you can give it a rake over and take off the dead white grass. The roots are still alive and will survive and bring the lawn back to life whether you do something or nothing at all. It's up to you.
The moths you see are probably what we in good old Blighty called Grass Moths or, to give them their Latin name, Crambus pascuella. They live on meadow grasses, clover and moss, so they are not altogether to be shunned as every lawn owner is keen to get rid of moss and clover, right? I have never heard of an American Sod Moth. But it sounds like a poor thing that has happened to wander into somebody's garden from the wildwood and has made enemies of people who live in a country that likes to shoot things.
Don't worry, things will sort themselves out. Sit in your lovely London sunshine (it rained here today), open a bottle of your favourite tipple, sit in a comfy chair on your nice dry grass and enjoy the show.