I'm looking for shapefiles of the British railway network from any time between 1900 and 1970 (before many of the railways were closed). There are many historical GIS sites (such as GBHGIS and Vision of Britain) but none of them seem to have any railway data.
You might find Open Railway Map useful. It is mainly focused on active railways, but also includes lines that are disused (i.e. the line is closed but the infrastructure remains) or abandoned (the infrastructure is gone but the route still discernable). It would be more difficult to find the routes of lines that have been completely obliterated, or which have been re-used for other purposes (as roads, tram routes, etc.).
Open Railway Map uses data from Open Street Map (OSM), so to get the routes you'll have to extract the data from OSM. There are lots of tools for getting data from OSM, but you might want to look at this Reddit thread for suggestions specific to extracting railway data. Also check if your GIS software includes a tool for extracting OSM data, e.g. if you're using R then I recommend the osmdata package.