I'm trying to understand women's work during the industrial revolution (like factory work) and how this was possible without contraception, synthetic breast milk to give to the child who requires breastfeeding, maternity leave, etc.
What did they do with respect to their children? They didn't have family planning services like we do today that can allow women to forgo their pregnancy until desired. Did they keep celibate? What else did they do?
Generally, once a woman had a baby, she worked from home doing odd jobs such as a seamstress. Babies need to be nursed less after 6 months and even less after a year (at which point they can be weaned). If a neighbor or family member was willing, they could watch the baby after mom returned to work (if she could get hired again- there was discrimination against mothers working because it was believed her husband should be providing for her family and employing her would take a job away from a man). As her children got older, they could watch their siblings while Mom worked, but generally by that point, a lot of women continued to stay home and do piece work and odd jobs. As children got even older (say 10 or 11), they would get jobs to support the family and mom would stay home to raise the younger children and still do odd jobs. Children also joined mom in piece work at home. This is for urban residents and company town workers. If you're a rural farmer, women just continued on with their their daily tasks while breastfeeding and caring for babies.
There were birth control options however. People mostly understood that a man ejaculating into a woman was what caused pregnancy, so you just had to block the sperm from entering the cervix. Or a man just doesn't ejaculate in his partner (aka, "the pull-out" method). Condoms existed in some form for centuries, using lambskin, pigskin, intestines, though they weren't widespread until the 1900's when rubber condoms we're mass produced. If you think about it, a condom and a sausage casing aren't fucntionally that different and people long ago caught onto that fact. Women had various types of barriers they could place in their vaginas during intercourse and have done so since ancient times. The proto-IUD was invented in the 1880's and various types of IUD's were developed throughout the 1910's and 20's. Women used spermicides and douched after intercourse in an attempt to kill sperm or flush it from their bodies.
If a woman did become pregnant, abortion was an option but it had to be done before the fetus "quickened" (you could feel the fetus move, so around 15 to 20 weeks). Prior to that time, it was believed the fetus wasn't alive, so removing it wasn't morally objectionable. Women have used abortificants since ancient times as well and by the period you're asking about, pills were sold in drug stores to induce abortions. An abortion could also be induced by shoving an object up the vagina and into the uterus to cause a miscarriage. It wasn't until the 1880's that laws began showing up outlawing abortion unless the mother's life was in danger. The Catholic Church also held the quickening view up until 1869 when it prohibited Catholics from engaging in abortion except to save the mother's life. In 1895, it did away with saving the mother's life and condemned all abortions entirely.