by Gaelan Steele, with help
This page attempts to visually represent the history of mainline railway passenger rolling stock in Great Britain, starting from the introduction of British Rail's TOPS classification system. I've also included London Underground rolling stock from a similar period, as the history of the two is in many cases intertwined.
Each box represents a class of train (or, in a few cases, two very similar classes); boxes are placed horizontally according to the year(s) in which they were built. Arrows between boxes indicate that one train was based on another, in some way; this can range from the design for one specific part (say, the bogies), to a modification of the same overall design, to direct conversions of existing trains. Icons under each train indicate the companies (or sectors of British Rail) that have operated them; you can hover over an icon to see the operator's name and dates of operation.
The nature of this format is that I can't capture everything; in some cases, a connection is present but there's no feasible way to draw an arrow. Within closely related families, the precise choice of where to draw the arrow is often somewhat arbitrary—it's not particularly meaningful to ask, for instance, whether the class 720 is "based on" the design of the 710 or the 345, as really they're all closely related iterations of the same design. I've also been somewhat capricious with the operator logos, in many cases omitting rebrands or leaving out distinctions between two operators continuing under the same brand; the intention here is to provide a rough indication of where a given train was used, not to precisely document the history of railway franchising. Please take everything here as a rough guide, not a definitive history. That being said, if you do think I've made the wrong call somewhere (or indeed am plain wrong!), do let me know!
Information was gathered in the first instance from Wikipedia, which (while imperfect) seems to have by far the most comprehensive freely-available history of rolling stock. This was supplemented with a variety of sources, including the excellent Blood and Custard and RAILCAR.co.uk. Much input was also provided by @Coastway_Will and @PacerClara.