Hi @umair,
thanks for pointing me in the right direction 
I got it now.
Running
$ bench doctor
-----Checking scheduler status-----
Scheduler disabled for www.example.com
Workers online: 3
-----None Jobs-----
revealed that for the one site, the scheduler was (for whatever reason) disabled. Using
$ bench --site www.example.com enable-scheduler
has enabled the scheduler and now the queue is sent. A useful resource is Unable To Run Scheduled Tasks - #4 by makarand_b