There is no curfew at LHR. There are limits on movements and noise quota at night, but these are applied by season (winter or summer*). So the lack of any night flights during the period of closure will effectively build up 'credits' in numbers of movements and noise quota, which can be used when flights are resumed.
As far as I can see, no special permission is needed to allow an increased number of night flights by permitted aircraft, though obviously there would need to be availability of ATC, ground handling and other resources. Surface transport would be a problem if a large number of flights arrived between 01:00 and 05:00.
* summer season is defined as the period during which British Summer Time operates, i.e. last Sunday in March to last Sunday in October.