Pain exists in the brain, not the body.
"If it seems strange that nerve signals coming from the back can represent vision, bear in mind that your own sense of vision is carried by nothing but millions of nerve signals that just happen to travel along different cables. Your brain is encased in absolute blackness in the vault of your skull. It doesn’t see anything. All it knows are these little signals and nothing else. And yet you perceive the world in shades of brightness and colors. Your brain is in the dark but your mind constructs the light." — David Eagleman in his book Incognito.
Without the nerves connecting your toe to your brain, you are unlikely to feel the otherwise agonizing pain of having had stubbed it against a coffee table.
Think of it as taking a microprocessor in a circuit and disconnecting every one of its pins except those providing power. How would that microprocessor behave? It's various I/O pins aren't gone, their state and response just became somewhat random.
We do not know that simply being severed from your body is simply experiencing no sensory input of any kind, and actually have a lot of reasons to expect that would not be the subjective experience for very long.
(Bizarrely, the dream pain has nothing to do with anything in real life that should have been pain. Apparently my subconscious is fine with dismemberment or stabbing or whatever, but what really bothers it is removing my orthodontic spacer from the roof of my mouth, which apparently it feels is me trying to remove the roof of my mouth. Subconsciouses are weird.)
[1]: https://www.britannica.com/science/phantom-limb-syndrome