"Mobilisation" has a wider meaning than just one event. Looking at reports from Keith, the process started for territorials with calling-up as early as 4th August 1914, nearly a week before the official proclamation. Since they already had a reporting structure those in towns would go to their drill hall, in rural areas to their platoon or company commander. Those rural units would then travel, often as a unit, to the headquarters town.
Once assembled there would be a period of drill, route marches, issuing kit, etc., but also free time in town for "fraternising". Reservists were also called up as part of mobilisation, but reported as individuals. "Call-up" got its more specific meaning of conscrption only later, when that began.
The second part of mobilisation was embodiment, which seems to go like this: Reservists needed to be allocated to a unit to serve in, but territorials were often reformed too. Their peacetime units might be too small, or their numbers might be needed to even up regular army units sizes. And that was just before he BEF was sent off - reinforcements in the field, whatever their units on arrival, were almost always used to plug gaps.
So exactly which part of that was the morning they were mobilised is not entirely clear.