Wet nursing
Someone else in the family would nurse the child or a wet nurse would be hired specifically for the job. A mother who recently lost a baby can continue to wet nurse another woman's baby. However, it isnt necessary for a woman to have given birth to elicit lactation. Adoptive moms can often use a breast pump to stimulate lactation in preparation for the new baby. The caveat being that they would stop ovulating - this is known as lactation amenorrhea. Relatives may also wet nurse to delay pregnancy as a form of birth control. Wet nursing was common practice in various cultures throughout history. Typically more common amongst the nobilty, aristocracy or upper classes. They weren't cheap and could also carry diseases like syphilis and HIV. It's still practiced today in parts of Asia and Africa. Back in 2008 the Chinese milk contamination crisis increased demand for wet nurses (hope this doesn't trigger the 20 year rule). Comparatively, the idea of wet nursing in Western societies elicits a cultural discomfort since it has associations with slavery. On the other hand we are okay with the idea of milk banks and even consider breast milk donation a noble cause.
Suckling animals:
Aside from the various characters from classical mythology who had been suckled by animals there were real instances of this method being deployed. The 18th century French used asses and goats. They were used at foundling homes and by mothers who were afraid their babies would catch syphilis from a wet nurse. Deborah Valenze in “Milk: A Local and Global History.” writes: "The cribs are arranged in a large room in 2 ranks. Each goat which comes to feed enters bleating and goes to hunt the infant which has been given it, pushes back the covering with its horns and straddles the crib to give such to the infant.” turn to page 177 for a diagram showing an ass suckling a baby. The Irish, Maltese and Swiss would use goats too. The English preferred asses. There were objections to these milks however. Goats milk has a distinct off putting odor. While asses milk apparently has a purgative effect (wonder if it is the osmolarity?). Asses milk is similar to human milk for its lactose, proteins, minerals, and omega-3 fatty acid content but has less fat. So often cream was added to asses' milk. Interestingly its gaining popularity in popularity again. A rather amusing fact is that people believed that children would take on the attributes of the creature that nursed them. A lioness would pass on courage while the Egyptians believed asses' milk would make the be stupid and stubborn. See Radbill's "The Role of Animals in Infant Feeding" for more of these ideas of character transference.
Early human milk substitute preparations:
Before formula people would use animal milk, typically raw cows milk mixed in with honey or sugar and other fats such as cream or butter. These early substitutes were woefully inadequate nutritionally. Contaminated ingredients and water could be a source of pathogens for preparations. It is a concern that still persists today with modern day formulas. The inadequacies of the milk substitute would lead to mineral and vitamin deficiencies. Also, the wrong concentrations of the various components used to make the feed would affect the osmolarity of the milk substitute and could cause diarrhea and dehydration. Again, its an issue that still persists.
Here's a fantastic paper to read about the development of infant formula over the last 100 years or so: Infant Feeding in the 20th Century: Formula and Beikost by Samuel J. Fomon. Department of Pediatrics, College of Medicine, University of Iowa, Iowa City, IA 52242
Edit: Clarity and sources