This is a great and interesting question and instead of giving you a confident answer, I'm going to explain why it's fairly complicated for those who don't specialize in this history to answer and then offer my best guess. I would, of course, be thrilled if an expert comes along and confirms or refutes my hunch but in the meantime, can share my recent experiences answering a very similar question about schools and colleges in the 19th century.

I was recently part of a project that included some work around the history of bathrooms on college campuses and I expected it to be fairly easy, mostly because I am incredibly spoiled by the documents and artifacts from New York State public education history; the state government sent out Commissioners beginning in the early 1800s who collected a great deal of data about the state of schools. It's because of those reports we know male teachers who worked during winter terms were typically paid twice, even 3 times, as much as woman teachers who worked the summer term, even in the same schoolhouse. We have a fairly rich accounting of the rise of free African schools in and around NYC, which textbooks schools used, and more importantly for the purpose of your question, counts of outhouses and their location near the school. (One of the key drivers behind these survey tours of schools across the state was to get more public tax dollars towards schools - so Commissioners would often highlight bad conditions, which included the lack of a privy at or near a schoolhouse.) Unfortunately, similar data weren't collected about colleges which complicated the research for this project. I likewise suspect such data are not available for cities.

It's possible infastruture census data are available for some cities, but if they are, it's probably from later in the century, after the introduction of public, flushing toilets and efforts to improve public sanitation. Instead of being able to see in a ledger the proximity between the places where people congregated and the places people took care of business we have to look for other evidence. But not necessarily evidence generated by people. We need evidence generated by men. Not because women are magical woodland creatures who never need to use the restroom or that men aren't people, but rather, because social norms on both sides of the pond were such that women, generally speaking, were more likely to stay closer to home than men and didn't have as much a need to use a restroom that wasn't their home chamberpot and/or outhouse. Which is to say, there were very likely no women working in offices in the 1800s, especially in the first half of the century. So, it's about men.^1

But it's difficult to know how men who might work in an office handled basic human needs as they were, I suspect, they're the type of men who wouldn't write about such things in the details we need to answer the question. I'm getting to the limits of what I can confidently speak to so I'll reel it in and offer that as part of the project I mentioned, I looked through a fair number of etiquette guides for young women heading off to or at college and didn't find any mentions - overt or covert - of how to politely handle the need to use the restroom. So one of the pieces of the historical record we can look to - etiquette guides - which offered guidance around aspirational behavior in public provides us little or no evidence around that particular behavior. Unless a modern day historian knows what to look for.

Not only would people rarely talk straight on about how they dealt with the call of nature, when they did, they often used different language to describe the space or equipment, depending again on class but also time and location. In this response about bathrooms on the Titantic, u/YourlocalTitanicguy provides an example of how the language has changed and can be misleading when read with a modern-day bent.

It's important to realize that the toilet and the bath were not the same thing and therefore not in the same room. Anything labeled "bath" is literally that, while "lav" would be used for toilet facilities (or in the cabins marked by a W for Washroom or WC).

When I look at school architecture guides from the early 1800s, I know that anything labeled "facilities" could be about any part of the building proper that's not connected to teaching (window design would often fall under facilities but desks and chairs typically wouldn't.) By the end of the century, "facilities" referred to the bathrooms, which may or may not be connected to the building. I came across the telling of a large building architect in the 1920s who wanted to label the room before the room with the women's toilets (often called a lounge or "power room") a "retiring room" but faced backlash as the word "retiring" was too close to how people described what was happening in the room with the toilets. /u/jbdyer does a great job here unpacking how history can get a bit twisted because of how people talk about or around bathrooms. In order to be able to explicitly answer your question, someone would need to know what to look for. That is, how people in a particular location or era talked about/around bathrooms and where they might talk about it.

And sometimes the location of the bathroom was implied - or we can speculate based on context clues. As an example, there are letters and journals from tutors at Harvard and Yale in the early 1800s complaining about the students' poor behavior. They spoke about the messes boys made and the foul smell in their rooms. Likewise, the school inspectors I described earlier used similar language when describing schoolhouses; they were ill-kept, with a noxious smell. That smell was very likely urine from boys relieving themselves against the side of the school but norms were such the men didn't come right out and describe it in detail.^2

All of which is to say, I suspect one of two things for those who worked in a building that was not their home in the first half of the 19th century, before water closets were common. They had a chamber pot and a screen or walled-off space in the office that the men used when they needed to use it or they simply went outside to use the nearest outhouse.

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1. It's worth stating explicitly that bathroom access was one of the reasons there were no women in offices. The presence - or absence - of bathrooms women could use without worrying about their social status was a determining factor in a number of situations, including whether or not a particular college was willing to admit women; no designated women's restrooms, no women students. (When Jeanette Rankin was elected to the House of Representatives in 1916, she had to leave the building to use a women's public restroom.)

2. Bathrooms really do play a fairly significant role in a whole bunch of history. Adding outhouses near a school was a fairly common first step in making the school more official and more importantly, ready for a woman schoolteacher. The feminization of the profession in American began in the 1820s and required a fairly dramatic shift in how schools looked; they went from unkept and dirty to clean and decorated -- from prison to parlor, as it were.

If you lived, like Charles Dickens, in London in the middle of the nineteenth century and were fortunate enough to work in an office equipped with the latest in sanitary technology, the odds are you’d be begging for the desk furthest from the toilet.

Suppose it is around 1850 and you work as a clerk at the Admiralty. Your office’s water closet—complete with a patented flush toilet from locksmith Joseph Bramah—still drains to a cesspool beneath the building. Though the fixture is located in a stairwell and equipped with an S-trap—invented by the watchmaker Alexander Cumming in 1775—noxious odors still escape into the workrooms. And, as you likely still subscribe to the prevailing miasmic theory of disease, you believe those foul vapors could give you a lethal illness.

In the fifty years since 1800, the population of London has trebled to three million. As the commercial and administrative activity of the capital surged and the population swelled, anxiety over water scarcity, quality, odor and pollution gripped the public and its democratic representatives in Parliament.[1] Successive cholera epidemics in 1831 and 1848 have killed 20,000 people in the city. Stormwater continues to send chemical runoff, human and animal waste, and carcasses and corpses through the brick sewers and into the River Thames, which is incidentally also the source of your unfiltered drinking water.

Sanitary reform is on everyone’s mind, including that of the novelist Charles Dickens. An editorial in his weekly newspaper Household Words decries the unreliable and unsafe provision of London's water by an inefficient network of private utility companies using a complicated and outdated system of pipes and drains.[2] Despite numerous recent public reports, including Edwin Chadwick’s “Report on the Sanitary Condition of the Labouring Population of Great Britain”, the supply of water is still expensive, intermittent and unpredictable.[3]

As you’re relatively wealthy from your work at the Admiralty, you have piped water in your home, even if it only runs from the tap for two hours a day and occasionally is cloudy or has a strange or unappealing taste. Thankfully, your office is equipped with similar comforts. The building where you work—the first purpose-built office building in Britain, dating from 1723—was equipped with the most advanced plumbing fixtures available, including Bramah’s flush toilets, when it was renovated about fifty years ago to provide additional offices and meeting rooms from which to administer the peripatetic Royal Navy as it patrols the Empire on behalf of Queen Victoria.[4]

The Admiralty was just the first example of a new building type—the office block—that has come to transform London in the past half century. Before its construction, government departments had occupied former royal palaces or aristocratic mansions, while commercial interests had used rooms in private dwellings or even coffeehouses—as had been the case with the insurer Lloyd’s—to conduct business. In the past fifty years, both Westminster and the City of London have seen the replacement of dense, dilapidated housing with monumental, neo-Classical office buildings.[5]

All of these tumultuous changes are recounted in the work of your favorite writer, Charles Dickens. Much of Dickens’ fiction is set in a familiar environment of business, law and government—from the frigid parsimony of Scrooge & Marley in A Christmas Carol to the circuitous chambers of the Chancery court in Bleak House to the tedious bureaucracy of the Circumlocution Office in Little Dorrit--and is populated by middle-class personages resembling yourself.

The people in Dickens’ stories, like you, live in a city in constant flux and beset by poverty, crime, filth and pestilence. Depending on economic circumstances and the good graces of their employers, the sanitary facilities in their fictional offices could range from a commode (a chair with concealed chamberpot), a privy (a seat with an opening positioned directly above a cesspool) or perhaps a water closet similar to yours. Modern public toilets have not yet been popularized, but there is still the option—at least for men—of relieving oneself outdoors against the side of the building.[7] You imagine the characters in Dickens' novels reserve this for when they wish to register a complaint.

Sources:

[1] Stephen Halliday. The Great Stink of London : Sir Joseph Bazalgette and the Cleansing of the Victorian Capital. Thrupp, Stroud, Gloucestershire : Sutton Pub., 1999.

[2] “The Troubled Water Question”, in Household Words: A Weekly Journal, Volume 1, No. 3 (13 April 1850).

[3] Edwin Chadwick “Report on the Sanitary Condition of the Labouring Population of Great Britain” London, Printed by W. Clowes and sons for H. M. Stationery off., 1843.

[4] John Summerson. The Architecture of Britain: 1530-1830. New Haven: Yale University Press, 1989.

[5] Michelle Allen. Cleansing the City: Sanitary Geographies in Victorian London. Athens: Ohio University Press, 2007.

[6] Rosemary Ashton. One Hot Summer: Dickens, Darwin, Disraeli, and the great stink of 1858. New Haven : Yale University Press, 2017.

[7] Lee Jackson. Dirty Old London: The Victorian Fight Against Filth. New Haven: Yale University Press, 2014.