There really was no specific age - it was not about hitting a specific milestone, but simply reaching a point where it became very unlikely that a woman would get married in the future. "Old maid" was a loaded term, not a neutral description in opposition to "wife". It was humorous - a joke.
"Maid" was, by the Victorian era, a rather old-fashioned way to refer to a young virgin woman, and the assumption was that almost every maid from the upper working class upward would eventually marry a man and stop being a maid; the two alternatives were that the maid would lose her virginity without being married (which would typically see her estranged from her family and needing to rely on sex work to support herself) or that she would never marry and therefore always be a maid. The former option was tragic, either in a realistic way or as the plot of a novel or melodrama, but the latter could go either way. On the one hand, it was also sad: by the beginning of the nineteenth century, it was generally accepted that every "natural" woman wanted to get married, have children, and make a warm and pleasant home for her family, which meant that an unmarried adult woman must have a hole in her heart, and perhaps a tragic story of her own involving a lost love. On the other ... there was something very funny to people about this female figure who completely failed to fit into the world around her. In failing this way, unless she was independently wealthy, she was dependent on friends and relatives and therefore was socially beneath them.
A short time since, I joined a party at the house of a friend. Amongst the number present were two ladies, whose peculiarities of countenance, and dress, and demeanour, bespoke their relation to that dishonoured part of the community usually designated "Old Maids". I soon perceived that little more attention was paid them than is absolutely required by ordinary politeness; our hostess had invited them from an obligation to do so; she certainly felt no desire for their company, for they were Old Maids, and of course must be disagreeable. Pained by the neglect they received, I took my seat by their side, and on entering into conversation with them, discovered that it is quite possible, however some may doubt it, to bear this unlucky cognomen and yet to combine sound sense and pleasing dispositions.
On my return home at the close of the evening, I could not abstain from the following reflections. How much is thought and said of Old Maids! How frequently are they grossly satirized! How commonly are they exposed to that merciless persecution called quizzing! Even children imitating their elders generally treat them with disrespect. Is a mischievous fun loving boy inclined to indulge his vicious propensity? Aunt Dorothy is sure to be selected as his victim: 'she is only an Old Maid', and may therefore be tormented with impunity. Should a thoughtless youth wish for amusement in sending anonymous letters, containing avowals of admiration and affection? Old Miss Rachael, or her equally aged and despised friend, Miss Griselda, are chosen to gratify his love of teasing and fondness for scribbling nonsense.
Nor is it by our sex exclusively such persons are misrepresented and annoyed; we may often observe the injustice and contempt with which they are assailed by their own. If it be remarked, that one of them is particularly neat in her person and attire, she is immediately and superciliously condemned as finical; or it is more than insinuated that she has some good reason for all this attention. If, on the other hand, she is not very nice in these respects, she is loaded with reproach, and stigmatized as a disgrace to her sex. If in company, she enter into conversation, it is said she is garrulous and satirical, while if she be silent, she is suspected of taking minutes of what is transpiring.
(From "Spinsters", Spirit and Manners of the Age: A Christian and Literary Miscellany, 1827 (extra paragraph breaks added by me for easier reading)
The only place I can find specific statistics is Women and Marriage in Nineteenth-Century England (Joan Perkin, 2002), which states that women's ages at first marriage ranged from 23 to 26, and men's from 25 to 30; generally, in poorer and more rural areas the average ages would be on the lower side, and among the wealthier and more urban, on the higher side. That doesn't mean that once they reached these ages, women would suddenly be considered past their peak, but as they aged farther and farther beyond them, it would become more and more obvious that they were not going to fill the societal role expected of them. A woman older than thirty or thirty-five would check the box for most people, as she would then begin to look "mature" in a way that, in the past, read as "past her peak", but from fiction it's clear that the stereotype tended to include a marked unfashionableness (hair worn pulled back tightly, clothes from a decade or longer ago) as well as physical unattractiveness beyond simply no longer being in one's early twenties (poor posture, a large nose, extreme thinness), and often personality flaws (harshness, indecision, silliness). The term "old maid" was very much a pejorative, which people would not generally use for someone they loved, and implied a lot more than just "you didn't happen to get married for one reason or another".