I'm asking specifically in the cases of meat and milk. Is the worst case simply an upset stomach?
The worst case scenario is death from complications due to bacterial food poisoning. The specific species' will vary by location. Here in the UK, food poisoning from meat and dairy products are most commonly due to Enterobacteriaceae (esp E coli), Campylobacter and Salmonella. Common symptoms include diarrhoea, vomiting, stomach cramps, elevated temperature, which if left untreated may lead to death due to organ failure from dehydration.
Note that eating spoiled food has the potential to cause illness, but may not do so. There is no way to know in advance. Also note that the species of bacteria that cause illness are often not the ones which cause products to be noticeably spoiled. Thus it it is possible to consume (visibly/smellably spoiled) food such as dairy (spoiled due to lactic acid produced by Lactobaccilus) but it is unikely that lactic acid or its by-products will make you sick (Lactobaccilus is included in many "probiotics")
Sources: Borch, Elisabeth, Marie-Louise Kant-Muermans, and Ylva Blixt. "Bacterial spoilage of meat and cured meat products." International journal of food microbiology 33.1 (1996): 103-120 https://www.ncbi.nlm.nih.gov/pubmed/8913812