This is a really fascinating question that is best served by first understanding the way women were treated during that time period, and why they would be sent to an asylum in the first place. I'm also writing this from the American perspective and I have no idea how to answer it from the non-American perspective. Also, heads up, my references are out of order because I added a couple after I did some proofreading.
In the 1800s, there was a lot of overlap between institutionalizing women (i.e. putting them in "a lunatic asylum") and, as the 1900s approached, incarcerating them. For example, Pouba and Tianen^(1) wrote that during the 19th century, "women were placed in mental institutions for behaving in ways that male society did not agree with" (p. 95). The process, according to that article, was that a woman would go to a doctor because of some "symptoms" that her family (usually, her husband) did not agree with. There, she would receive a diagnosis and ultimately get institutionalized. It was very easy for women to get institutionalized at the time - it's not like today where there's a lot of "red tape" involved in putting someone on a mental health hold or sent to a community mental health facility. Back then, the diagnosis would be something along the lines of "insanity by way of nymphomania" (where "nymphomania" refers to a manic condition of desiring/having too much sex) or "insanity by way of suppressed menstruation" - women could literally laugh too much or express too much interest in sex and a "well-meaning" family member could bring them to a doctor to have them institutionalized.^(4) I mean seriously, look at the table on reference #4 on p. 141 - it's a study looking at women who were institutionalized in the late 1800s and all the reasons for being admitted - out of 567, 70 were admitted for "irritability towards the family," 17 were admitted for "masturbation" and 2 were admitted for "refusal to have sexual relations with spouse."
So anyway. Here's the thing. Moral deviance was increasingly being seen as, really, a form of crime. When the first women's prisons started opening up in the mid to late 1800s, women could be incarcerated for those reasons listed above as well.^(2) Now, I don't mean to imply that women were commonly incarcerated - even today, there are far fewer women's prisons than men's prisons. I also am not using the word "criminal" to mean someone violent or someone who steals things (although women could certainly be incarcerated if they did those things). In fact, many criminologists (including criminologists from almost a hundred years ago!^(3)) will argue that a "criminal" is just someone who violates a social norm. Today, one need only look as far as drug laws to see how we are currently moving to re-define "morality" in terms of drug use. So what was happening in the mid-1800s was women were being put in asylums for behavior that was outside of the norm and at some point, society collectively began to decide that behaving in this way was not just abnormal, but it was also actually a crime for which women could and should be punished. In reference #2, the authors write:
we discovered that an acclaimed doctor who cared for the women and girls at the [first women's prison, Mount Pleasant] from 1873 to 1883 advocated female circumcision and removal of women’s ovaries to cure nymphomania and masturbation.
Now that I've given you a little bit of historical context for what women were dealing with and why they could be institutionalized and/or incarcerated at the time, I want to try to address your actual question.
What happened in a lunatic asylum during the 19Th century when a patient became pregnant?
What you wrote implies that the patient became pregnant while in the asylum. If this is the case, well, to be honest, these events are not well-documented. Below is the best I could come up with by combining the historical context and a few sources.
If a women became pregnant while in an asylum, it would imply one of a few things: a) the woman was impregnated by a male staff member; b) the woman was impregnated by another patient; or c) the woman was impregnated by a visitor. Furthermore, there are really only a few different options for what she could do with it -- abortions were not really safe or 100% effective at the time, and having an abortion could be described as "morally ambiguous" at best (that's a story for another time). Furthermore, carrying to term may or may not be an option because women were subject to all kinds of horrific abuses while they were institutionalized (which could lead to unintentional miscarriages or the death of the woman). You can read "Ten Days in a Madhouse" for the account of a journalist who pretended to be insane so she could be institutionalized - the "quality of life" in some of those asylums was horrific. If you've ever seen American Horror Story, this is where that one season got its inspiration (except that one particular season doesn't make asylums look quite as bad as they were).
In prisons, women could actually be punished for becoming pregnant because it implied that they were engaging in sexual relations when they shouldn't have been (at that point in history, "rape" wasn't really something to be considered). In Auburn Prison (NY, 1825), women were simply housed in a segregated unit of the men's prison. A woman got pregnant while in the men's prison (and the implication from some documents is that it was due to sexual assault by an inmate), she received lashings for it, and some scholars argue that this is one of the main reasons why women-only prisons were first opened (so that women could be totally separated from men so as not to tempt the poor men into inappropriate sexual relations).^(5) Other accounts say that a woman who was raped at a separate institution died in childbirth, leading to the first women's prison being opened. This implies that she did not receive adequate care for her pregnancy, or perhaps that the guards did not even realize she was pregnant (or did not want to admit that she was pregnant).
The long story short is, if a woman became pregnant in an asylum, she would probably "try" to carry to term, but she wouldn't receive adequate care and/or she would be blatantly abused. Thus, either she or the fetus/child (or both) would die. If it was a prison rather than an asylum, she would likely be punished, but would still not receive adequate care and she or the fetus/child would die. I'll close with a narrative from a dissertation^(6) I found while researching your question:
After her release from the asylum, Lathrop continued to fight for her cause in exposing the abuses in asylums, especially in Utica where she spent two years of her life. [...] Lathrop asserted that she was drugged and sexually abused by male attendants. She said, "I was at the mercy of unscrupulous and vicious men, with no one to appeal to for help or redress, with no escape, no refuge, powerless to protect myself." She was not the only one who knew what went on with the attendants. [...] Further, she stated, "I was also cognizant of the fact that a patient who had been confined in the asylum for about two years had given birth to a child [while in the asylum]." That this woman had a child was something unhidden from others, "this fact was generally known to the attendants and patients alike in the asylum." The child did not live and a patient saw the dead body as it was taken away. [...] Lathrop declared, "I believe that few patients escape the wanton lust of the physicians."
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References formatted half-assedly
1 Pouba, Katherine and Tianen, Ashley (2006). Lunacy in the 19th century: Women's admission to asylums in the United States of America.
3 Fuller, Richard (1942). Morals and the criminal law.
4 Warsh (1988). The first Mrs. Rochester: Wrongful confinement, social redundancy, and commitment to the private asylum, 1883-1923.
5 Rafter, Nicole (2009). Partial Justice: Women, Prisons, and Social Control. Chapter 1: "Much and Unfortunately Neglected": Women in Early and Mid-Nineteenth Century Prisons.
6 Walter (2011). Insanity, rhetoric, and women: Nineteenth-century women's asylum narratives.