An interesting point about this. AmE is far more conservative than BrE, and we have retained the Teutonic tendency to roll numerous smaller nouns into larger ones -- even more so than BrE.
This is revealed by intonation. Both "weekend" and "new year" are stressed the same as "bookcase" or "balljoint" in AmE.
In BrE, the first two are intoned rather more like separate words should be: week END, new YEAR.
(You will remember that the Anglo-Saxon tradition still alive in English is generally to pronounce multi-syllabic nouns with the stress on the first syllable.)
Perhaps we should, in AmE, start writing "newyear" as one word, as it is (I believe) in most Germanic languages.
So, "at the week END" is intoned in BrE rather the same way we would in AmE if we used the much much less frequent, but still plausible, construction: "at the week's end."
This should explain the difference in usage between at and on: we say, on Saturday, and on Sunday, so if the weekend is uttered as a Teutonic compound noun, fusing the two days into one concept, on the weekend is very natural.
If, on the other hand, you think of it as "the week END", it should be introduced by the preposition "at": At the week END" as we would say "at the end of the week," even in AmE.
