I can't add much detail about closures in the North-East, but have written elsewhere here about those radiating from Taunton, where, in the mid-1950s, there were local services on six lines, three to the west and three to the east, which all progressively got closed over the next 15 years, only main line service remaining.
Basically, the local trains, all of them, had pretty much ceased to be relevant for the local population already. Take the Barnstaple line. This ran out through two small towns, Milverton and Wiveliscombe, then through unpopulated country of Exmoor until it reaches another small town, South Molton, and then runs down to Barnstaple. It's pretty lengthy as country lines go, 45 miles, and took two hours to traverse. Setting aside Summer Saturdays, there had long been about 6 trains per day.
Now the small towns mentioned had, by no coincidence, established bus services. The bus from Taunton to Wiveliscombe ran every 30 minutes, 7 days a week. Furthermore, at every point the bus ran through the main street. The railway at Milverton, and especially Wiveliscombe, was well removed from the centre, and this pretty much applies at Taunton as well. Therefore the railway had lost what relevance it once had there after about 1928 when the buses became established. The same applied from South Molton into Barnstaple. The long bit in the middle had no bus, but nobody (railway staff apart) really lived there. By the time, which I just recall, that Beeching was around, the standard 43xx and three ex-GW corridor coaches had given way to a two car dmu. Both cars powered, so four bus engines running. If the passengers in the train, still half a dozen a day, at any point were into double figures that was unusual. Nevertheless there were two shifts of station staff and signalmen at all points along the way.
I think everyone we knew in those times who lived out that way had bought a car. Farmers certainly did, for their own purposes for which any other means would be impractical, such as visiting other farms. And those who didn't, like impoverished farm labourers, generally didn't travel, to an extent that does not happen now. The railway had become irrelevant long before. The one social impact noticeable was the impact on railway employment. It was said the railway had been the biggest employer in Taunton. A great deal of that went, with economies on the main line as well as those closed.
If you are interested in Beeching, another book to find is Stewart Joy's "The Train that Ran Away". Joy was Beeching's chief economist, a very practical Australian, who dug out all sorts of nonsenses with the way information was presented. He found for example an early morning local train on the ECML from Dunbar to Edinburgh, for which a main line loco ran out from Edinburgh, which reported 60 passengers a day. When he inspected this train, he found that all but a couple of these were railwaymen coming into Edinburgh, on free passes ...