On the one hand the potential economic impact in the long term of having that railway in place is quite large, but the reality is that as proposed it just wasn't going to do all that much and quite likely wouldn't have stayed intact long enough to have that impact. Remember that there never really was a proposal for a single Cape to Cairo, it was really a project to stitch together a series of routes with multiple gauge breaks and no serious prospect of operational integration. Financially there really never was a case for the railways, in terms of colonial economics routes to the sea will always outperform the overland route and while it could save time for passengers there just wasn't (and still isn't really) the kind of massive levels of north south passenger traffic that could pay for a project on this scale. Doubly so when you consider that Cook is exactly right; by the time there is any realistic prospect of completing the line aviation is at least on the horizon, and very likely essentially a reality.
Realistically I would expect that the network as a whole would never have been anything like financially viable and that any semblance of an integrated network would have fallen apart as Africa was decolonized. If anything it would be taken as a sign of the folly of empire and accelerate Britain's disillusionment with it.
That said, even more so than OTL it would probably be seen as the great lost opportunity of Africa, and there is every chance that we would be seeing a variety of initiatives to rebuild and reopen missing links around now. You might well see it connected very closely to the various things that make of the Trans Asian Railway initiative. IMO to see a major historical impact you would really need the Cape to Cairo route to be at least mostly a single gauge and COMPLETED (rather than just barely started) in the 19th century. While that would approach ASB (at least without a massively different conception of empire) you then do start to get potential for very different economic condition and relations through most of the continent. Even then its not going to make financial sense, but at least it starts to get the potential to be a long lasting system.
All that said, let me throw a bit of insane imperial rambling out there: how about a contiguous overland British Empire stretching from Singapore to Cape Town with a railway network connecting it all (when you include post WWI African and Middle Eastern acquisitions that "only" actually requires conquering Persia and Siam). Perhaps it could be just in the realm of possibility if the railway manias somehow had an imperial bent to them.