As I remember, the answer to whether disruption was handled better in BR days compared with now is "yes and no".
"No" in the sense that information technology was poor then - no internet, apps, p-i-s etc. The main way of getting a message to the passenger was through the signalman stopping the train, giving a message to the guard who walked through the train and passed it on to the passengers and to the porter at the next station, who chalked it up on a blackboard near the ticket barrier.
"Yes" in the sense that fewer trains, more alternative tracks and greater route knowledge of staff working in a nationalised industry meant that services could be diverted more easily and quickly via other routes, obviating the need for rail replacement buses, etc.
As has been mentioned above, staff were also ready to offer unofficial remedies to ensure passengers got to their destinations. My Father told of missing the last train from Manchester to Halifax whilst he was returning home on leave. The next train ran only to Todmorden, and after depositing the coaches into the siding there, the driver offered him a lift on the engine from there to Sowerby Bridge, where it was going on shed, meaning he had a short walk rather than a long one. On another occasion when I was travelling from Halifax to Aberystwyth for an interview at Lampeter College, the train was so late from Shrewsbury into Aberystwyth that I missed the last bus to Lampeter, and a kindly railway porter took me in his car. Looking back, I don't think he had an ulterior motive other than wanting to help, but such help would not be offered now, nor would you accept it if it were.