And when the trains were late....
For internal BR "services" it didn’t matter as much. Urgent items were put on the next available train. So if the trains were running late or out of sequence, whatever was the next service is the train that was used. And either you were informed at what time the train was expected (and at which platform) or you contacted the relevant person and found out yourself. In my case, we got our information from the PSB.
If the item was a spare part to deal with say, a signalling failure that was causing massive disruption, operations were advised. They then may arrange that that particular train was given priority if needed. Very occasionally, the driver would be instructed to stop at a particular signal box, crossing keepers box, signal or junction and the spare part would be passed to the team on site.
Oh, and also very occasionally, a train would be used to transport a small team (of two or three people) directly to site to deal with a signalling failure.
For non-urgent items, again, it wasn’t an issue.
And most of the time, this method was faster than a man with a van driving to go and collect said item. Which is what happened once the TOCs were no longer part of BR. Except it was worse, as often an entire two or three person team would have to go.