Just to add,

I am looking for flights out of Brisbane to Heathrow on 3 or 4 June. The EK site is showing no availability across all classes on all flights.

Do I assume this is a website problem or is there festival I am unaware of going on. It is a bit early for the returns to be booked out by returning supporters from the Lions Rugby tour as that hasn't started in OZ by then!

PS. Interestingly, to answer my own question, the same flights are available via Qantas so perhaps it is the site.