In my experience it's common to be able to do an interline bag check even when the tickets are separate. Some of the low-cost lines may not allow this, but I wouldn't expect a problem with BA and Alitalia. In my distant memories of connecting in Heathrow, I think there were security checks, potentially with long lines, both when leaving the arrival terminal and entering the departure terminal, so it would take more than 15 minutes. Just hope your incoming flight is on time and be aware that there are several London-Rome flights a day if you miss your connection.

Andrew Missouri