"Whale" is standard in this kind of context. There is another question about it with some more information: Using "whale" as a verb
A side note: I have seen "on" used more often than "upon" in this expression. "Away" is also possible. Apparently, it can also be used as a transitive verb with a direct object.
Unfortunately, as Jim says, the spelling you have been accustomed to use ("wail") seems the least defensible from a prescriptive point of view. It has been noted in an entry on the eggcorn database: whale » wail.
The OED entry for whale, v.2 provides the following information:
Of obscure origin. Commonly regarded as a spelling of wale v.1, but there are difficulties of form, chronology, and meaning. Perhaps originally = to thrash with a whalebone whip (see whalebone n. 3b).
Now U.S. colloq.
- trans. To beat, flog, thrash.
1790 F. Grose Provinc. Gloss. (ed. 2) Whale, to beat with a horsewhip or pliant stick.
- transf. intr. To do something implied by the context continuously or vehemently.
a1852 F. M. Whitcher Widow Bedott Papers (1883) vi. 67 You remember that one that come round a spell ago a whalin' away about human rights.
So the primary spelling seems to be whale, but it seems there could be an etymological argument for spelling it wale (if it is connected to the word wale, meaning a raised line or ridge, the source of the word weal). That said, I haven't seem any evidence indicating that the spelling wale is used in practice in any carefully edited works.
It doesn't seem to show up on the Google Ngram Viewer:
(I assume "wail away at" and "wail him" show up because of the other, more legitimate uses of the spelling "wail".)