Congratulations, I’m CACKLING at this. Omg, the only way this could be better is if a ghost pops up and Nightingale has to deal with it and tour-guide!Peter is VERY suspicious/intrigued.
I’ll go you one better: it turns out that Nightingale is in fact the embodiment of that post about immortals who weren’t really paying attention to major historical events and the fact he corrected Peter on is the one thing he knows that Peter doesn’t.
Peter spends the next few months trying to interrogate him on various things and being told that Nightingale a) was in Sri Lanka b) remembers things but not the thing Peter wants to know and/or c) just wasn’t paying attention. Peter is only slightly mollified by being introduced to some of the older Rivers, because half of them weren’t paying attention either, especially to things that didn’t affect local hydrology.
(I am 90% sure this is true in actual canon but it’s even funnier if knowing historical facts about London is Peter’s job and not just his hobby)