Well, you might be able to find a public toilet or stop somewhere. Your wife, however, is going to have a harder time. This is broadly because of concepts of who is and isn’t supposed to be in public space, which, for the Victorians, was guided by the idea of “separate spheres.” How this ideology functioned is complex, and subject to a lot of discussion among historians, but broadly speaking separate spheres was the idea that the public sphere was for men, and the private sphere was for women. This ideology emerged from the middle class, but became a dominant force in British society at large over the nineteenth century. (Davidoff & Hall, Family Fortunes, 2007)

What this means, functionally, is that there aren’t that many public spaces in London that would be open to “respectable” middle-class and aristocratic women in the nineteenth century. So, if you’re out on a walk, your wife is not going to be able to go into a cafe or bar to use the restroom there. In fact, your wife might find it difficult to even find a place to get food or drink. Because of this women’s social lives are primarily constrained to private circles: they visit people in their homes, dine at home, etc. and they could only travel as far as they had family or friends en route. Middle-class men have slightly more options, in addition to their workplace, they could also go to their club, which was a space specifically designed to accommodate spending time away from home.

What about public bathrooms? The first public toilets in London opened as part of the Great Exhibition of 1851 in Crystal Palace; they were in some ways an attraction unto themselves because they were a novelty. Part of the reason they existed was specific to the exhibition: the exhibition was a huge event, visited by mixed groups, who were often there for the whole day. In order to encourage attendance, a bathroom was a necessity. After this, more public restrooms start appearing for men as the century progresses, the concept of public restrooms for women remains controversial. (Penner, Barbara. "A World of Unmentionable Suffering: Women's Public Conveniences in Victorian London." Journal of Design History)

One of the first places women’s restrooms (and tea rooms) start to show up are commercial spaces: department stores, theaters, hotels, etc. Many of these had an entirely commercial motivation; if women could eat and use the bathroom in a store, they would probably stay longer and spend more money. These spaces were not uncontroversial, as some argued that allowing women to socialize in public in this way would lead to corrupted morals or uncontrolled behavior. (Rappaport, Shopping for Pleasure, 2000)

Through the early twentieth century, as women started to enter the workforce, a similar issue starts to play out in office and factory spaces as well, since many of those spaces were constructed on the premise that women would not (or should not) be in them long enough to need them. Even today, you’ll see comments about, for example, older university buildings not having equal numbers of men's and women's restrooms.

This CityLab article is about the U.S., not London, but describes a similar phenomenon: https://www.bloomberg.com/news/features/2018-12-03/the-rise-and-fall-of-the-women-s-restroom-lounge

Also apparently the Great Exhibition toilets were recently found: https://www.royalparks.org.uk/media-centre/press-releases/toilet-remains-from-spend-a-penny-exhibition-uncovered-in-hyde-park#:~:text=The%20remains%20of%20a%20165,former%20Great%20Exhibition%20of%201851.