A couple of suggestions:
Firstly,
BT Phonebook lists two entries for Handpicked Collection in SW8: the 0845 number and (020) 7720 9224.
Secondly, Googling the press number lead me to:
http://www.funpicked.com/delivery-terms-and-conditions/This is another website they run with the same 0845 number on, in which they invite customers to ring the number given for press enquiries.
The Internet Archive stores websites so we can see how they used to be. In October 2007 Handpickedcollection gave the phone number 020 7501 1057 and e-mail address john@handpickedcollection.com (for John Glenn) on the homepage:
http://web.archive.org/web/20071023060157/http://www.handpickedcollection.com/This is the same phone number as for the press office.
According to the website, the 0845 number is open 24/7. I would be very surprised if this is not outsourced. If it's a relatively small operation, then it wouldn't make sense for them to pay staff to be in the office at all times.
In which case, we really need to work out where calls are answered.