Under the present LHR rwy arrangements, simultaneous landings do take place in the early morning 0445-0600. Granted it is not busy then.
Not quite.
Overnight, until 0600, LHR operates a night alternation schedule which involves use of
one runway only.
It's from 0600 to 0700 that
both runways are typically used for landings (known as "TEAM" - Tactically Enhanced Arrival Measures).