Not a historian, but this is an excellent question to use the Google Ngram viewer. This is a search engine which looks for ngrams (combinations of words) within book corpora Google have scanned. I've found it a fascinating tool for looking up all sorts of common and uncommon words and phrases -- it's often quite surprising which are novel and which have been around forever.

Though it turns out that the answer isn't particularly straightforward, at least for published English. Looking at the British English corpus from 1500 to 2008 we see both forms in use from the early days ... and comprising a surprisingly large percentage of all written words. Note that the Ngram viewer dispenses with dashes between words, so "two and twenty" is equivalent to "two-and-twenty".

Closing up our time window to 1800 - 2008, we see "two and twenty" being more popular for much of the period, and the cross-over happening quite late -- sometime in the 1970s! Bumping up the interval once more to 1960 - 2008, I'd say the evidence points to 1974.

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Update: As /u/skirlhutsenreiter notes below, the Ngram viewer does distinguish between "twenty two" and "twenty-two". Adding this to the diagram we see a crossover date of roughly 1738, which seems rather more appropriate.

Answering those who question the method: the Ngram viewer is based on, pretty much, the raw substance of history: the written word. It's what makes it so powerful. Yes, you can misuse the tool (as I did inadvertently), but by examining the results and digging into the source documents (Google generally presents at least a brief passage, and often the whole work for anything out of copyright), you can refine the analysis.