A Google search - nay, several Google searches - found little direct evidence of this. Best I could find was Nature's Calendar, a site from the Woodland trust for the UK Phenology Network; the page pointed to defaults to plants, but you can select birds and then cuckoos, and see reports of first sighting by date on the UK map. You can even compare different years; I compared 2008 with 1998; several times more cuckoo reports in the later years, but maybe this is because 1998 was the earliest year and their network got better entrenched since then. Nature's Calendar also reports average sightings by year, which did show a later tendency, but usually such averages are suspect if you can't see the underlying data. Anyway, a far cry from my hope of a hundred year proxy database!
But then shock horror, I discovered the underlying premise is false; the Times has not been publishing first cuckoo letters since 1940 or so. They do occasionally publish something, eg:
'On April 21, 1972, Mr Wadham Sutton got away with a letter to the Editor reading: “Sir, Today I heard a performance of Delius’s On Hearing the First Cuckoo in Spring. This is a record.”'Bah, humbug!