At present it seems that the Task History Start Time is set at the same time as the task is created, and therefore is the same as the creation time. Is there any possibility in Activiti to allow a user to signal the start of a manual task and hence the start time would be when the task is actually started and not when it is created?
Not directly, but you can assign to a group and have the user 'take' this task. Then use a tasklistener which listens for an assignment event and write something to a database only when assignment to an idividual is done.