Unfortunately, there could be multiple causes for this issue you are seeing. Without seeing the inside of the wall or the configuration of the waste/soil lines it might be impossible to give a definitive answer.
It is very likely the leak is always occurring when the water is running, it is just noticeable once the wall material become saturated to the point it cannot hold any more water, and then begins leaking into the room below.
I think is unlikely the drain pan for the shower is the culprit if the toilet is actually affecting the rate of water intrusion. The toilet would have to push enough water up the shower drain to fill the trap and flow into the pan. If it could do that, then you have another problem further down the soil line.
Given your description these are the places I would inspect:
Check for blockages. Even if a blockage is the culprit, there is still a problem. Ideally the overflow should appear in your fixtures first, not hidden behind the wall.
I would check the joints in the stack. Typically, a toilet soil line is separate from the waste line of a sink or shower until the soil stack, where they collect and run to the sewer. If a joint at the stack has failed, the waste water flowing through the fittings is seeping through the joint and down outside of the stack to the cut-out in the plate between the floors and then into the ceiling.
If the toilet and shower are sharing a drain line check the joints in this line too. Pay attention to the traps, while installing a new drain pan or toilet if too much force was applied it could have damaged the joints of either the shower or toilet. E.g. remodeled floor changes the rise of the toilet flange above the floor; new toilet is installed and tightened down, pushing against the soil line, stressing the joints and forming a leak. Cast iron is not forgiving of this abuse. Also check that the pipe is being supported at regular interval so it is not sagging and causing unnecessary stress to the joints.
Check that the shared line is sized for the flow, is sloped correctly, and doesn't run too far. If any of these factors are incorrect, air-locking can occur. This can decrease the amount flow the pipe can handle, which leads to water filling or backing up in the line until it breaches, say the toilet flange, where it flows over the flange edge and down the pipe into the space below, all hidden by the toilet pedestal itself. Your pipe should not have a slope of greater than 0.25" per 12" run. The pipes size from the toilet should be at least a 3" pipe, perhaps bigger if the shower is joining it; a pipe of such duty should not run more than 6' to the soil stack.
