How to calculate scale

Your calculations appear to be sound. I suspect the problem is with the very first assumption: "the imagery is supposed to be to a scale of 1:250000" (emphasis mine).

I know nothing of JOG-A or CADRG but I do know that, when it comes to displaying images on a computer screen, there are often many scale transformations that take place between the original image creation and the final display on a computer screen. There can be so many scale changes that an original statement of scale can appear to be almost meaningless in a later context.

In the case of analog photogrammetry – I use this because at least the film (the negative) had a fixed physical size – the imagery is given a nominal scale relating measurements on the original photographic contact print (inches, say) and measurements on the ground (e.g., miles). Even if the images have been enlarged or reduced, for reasons of convenience, the original statement of scale is often carried with the imagery as part of the meta data, to give folks an idea of how much detail is captured. Ideally, there would be notes that the statement refers to the original capture and not to any current reduction/enlargement, but these may be missing.

As soon as you move into the digital realm, the display scale can so easily be changed – intentionally by users/designers/publishers or automatically by software (to make things fit the page/screen) – it soon becomes very difficult to reconcile nominal statements of scale with actual physical measures of scale. That is why "bar scales", or graphical scales that are part of the map or image, are so useful.