With our not being restricted to Great Britain and / or UK: I'd choose a day -- ordinary weekday, or Saturday if that gave better opportunities -- in "north-central / west" Ireland, both sides of the border, in summer 1957 (I was then actually aged nine; didn't in fact get to Ireland in any way or shape, for another dozen years -- but from an early age I read about, and yearned for, the stuff "on the menu" for my chosen 24 hours.). Said 24-hour "bash" would be in the greatest part, a last look at things which were very shortly to vanish: the Great Northern Railway of Ireland's secondary-line system centring on Clones, and -- connecting therewith -- the cross-border independent Sligo, Leitrim & Northern Counties Railway -- all of which lost passenger services, and a large proportion of the whole being totally abandoned, with effect from the end of Sep. 1957. I'd also include as much as circumstances would permit, of CIE's 3ft. gauge, all-steam, Cavan & Leitrim system: which would be abandoned a year and a half thence.
I don't have the relevant timetables, so must use guesswork as to what would be possible in a long day's travelling. Re lines covered, I see the exercise as very probably having to be "quality rather than quantity": the schedules of non-Great Northern lines in the Republic, tended to be meagre. I'd see covering the following itinerary either in the direction set out; or the opposite direction; depending on which would work better and give, relatively, more coverage. (In extremis, some resorting to buses or taxis might have to be done to get between desired goals; the thought of having for the trip, a confederate with a car -- to allow via "mixing-and-matching", overall maximum rail-travel variety -- is tempting: but that is not really my preferred way of doing things, and would feel a bit too much like cheating.)
In the sequence which I'm about to describe: starting out early morning on the Cavan & Leitrim system, and travelling over as much of the same, as schedules would permit in the overall course of the day's doings. This would build in if at all possible, travel on a C. & L. passenger or mixed train from Ballinamore to Belturbet; changing there, to the Great Northern broad gauge. I understand that passenger on these GNR(I) lines at this time, was a mixture of steam and railmotor -- what with all that was hoped to be done, type of power on my workings would have to be whatever it turned out to be. On the GNR(I): Belturbet -- Ballyhaise (which branch I suspect was all-steam; could be wrong there) -- on to the big junction of Clones; from there, over the border into N.I., on the Great Northern's Clones - Enniskillen -- Omagh line, as far as Fintona Junction. Alight there to travel over the very short branch to Fintona, worked by the unique double-decker horse tram; back to the Junction, and retrace steps to Enniskillen. From there, a Sligo, Leitrim & Northern Counties Railway catering-for-passengers working back into the Republic, and finally to Sligo (running powers over the CIE main line, for the last stretch after Collooney).
Have planned the journey in the direction as above, for preference: because in the S. L. & N. C. R.'s last years, all its passenger workings were railcar or railbus, save for a mixed train, steam of course, which left Enniskillen for Sligo, late afternoon / early evening (no corresponding mixed working in the opposite direction). The ideal situation would be, to finish the day's journey by travelling on this train for the length of its run. However -- if doing the undertaking the other way around, would yield better trackage-covering opportunities overall; I'd do it the other way around -- Sligo to Enniskillen, earliest working of the day: railmotor as it would be; finishing the day on the Cavan & Leitrim, as far as it might be possible to penetrate to on same (Arigna? Dromod? at worst, perhaps just Ballinamore...). With Irish weather being as it renownedly is: I'd appreciate being able to order a guaranteed beautiful sunny day for the bash.