When death entered into the world with Adam and Eve's sin, there had to be some way to keep the world clean and sanitary, so bacteria, flies, worms, mice, rats, buzzards, hyenas, etc. would eat dead things, so I would say right away when death entered the world that animals, birds, insects, and microbes started eating meat.
When God killed the animal to clothe Adam and Eve, He would have also shown them how to kill the animal humanely, prepare it's skin/hide, and prepare and eat it, because otherwise it would have been wasted, and animals would have suffered while Adam and Eve fumbled around figuring out how to kill them for whatever reason(s) needed. Unless the carnivores, like tigers, lions, small cats, bears, etc. were scavengers and not hunters starting at the moment death entered the world, and the meat from animals slaughtered for clothing and sacrifices was given/left for them, then I think humans also started eating meat starting when God sacrificed the animal for them. God is good, so even when dealing with sin and its consequences, He would have done what was right by His creation.
It doesn't make sense that people would be keepers of sheep and keepers of cattle and use them for skins, fur, clothing, and offerings alone. Abel brought of the firstlings of his flocks and fat, so what happened to the other meat? God also punished Cain by saying the ground wouldn't yield its strength to him when he tilled it, so if he had no crops, he would have had to eat something, which I think implies meat. It could also mean he had to trade or steal, which might be at least part of the meaning of God telling him if he didn't do well, sin was at the door.
Animals, which God intended to eat meat, have sharp, pointy teeth and/or beaks and/or claws/talons. Something to tear, rip, etc.
Human beings have some sharp, pointy teeth, but also flat and grinding types, so that implies God's design to eat a variety of foods.
The Bible doesn't tell us how long it was before Adam and Eve sinned, but I bet the devil worked fast, and maybe took advantage of God resting on that first Sabbath and did his dirty work then. Adam and Eve might have been less than one day old when they sinned. Given that some animals, such as felines, are obligate carnivores, and some insects and birds can't go very long without eating meat, I don't believe it was very long, because God wouldn't have wanted animals, etc. to suffer, and He knew what was going to happen and when.