Often when they don't turn up for a scheduled appointment. For old people that's quite often something like a church coffee morning, U3A meeting etc. As you can imagine this stopped being the case during covid, so many people ended up going a lot longer before they were found. Milkman noticing milk not being collected from the doorstep is another classic one. The smell is actually a legitimate answer unfortunately - you do hear stories of people reporting a terrible smell which turns out to be a dead body.
More distressing than dead people not being found for weeks is people that have a fall or some kind of medical episode and aren't found for days. A family member knows someone through a church group who had a stroke at home, and fell down next to the radiator beside his bed, couldn't get up. When people noticed he was missing a few days later he was not only very ill from the stroke and being unable to eat or drink etc for days, but the whole side of his face next to the radiator was burned, luckily he survived.
My next door neighbour had some kind of fall or episode in 2020 (not really appropriate to ask), not sure exactly but was lying on his bathroom floor for at least two days (probably three) before the milkman noticed he hadn't collected the milk and asked neighbours if they'd seen him. Someone got into his back garden and found the back door unlocked, and after finding him called the ambulance and a few of us helped clear a path so the ambulance guys would be able to get him out faster (turns out he was a little bit of a hoarder). Doubt he has any memory of me being in his house as he was pretty out of it, but I'll always remember that.
My grandma had a fall in her garden and couldn't get up having broken her leg - luckily neighbours saw her / heard her and called for help, and kept her warm while they waited for an ambulance (which took about 6 hours to arrive, so good job they were patient).
Really sad situations to think about. Sometimes forget how vulnerable old people living on their own are. If you know an OAP living on their own try and convince them to get an alarm button to carry round with them or something and if possible some kind of regular visitor like a cleaner or carer.