Well the best real life comparisons we have to a less centralized nervous system are cephalopods (focusing on the octopus) and arthropods (other animals also have less centralized nerve distribution, like worms and jellyfish, but I don't find them good examples for a sentient creature).
So let's start with what we know about these creatures. Octopuses might be the best arrangement according to what I understood you want. They do have a centralized brain, but about 3/5 to 2/3 of all of their nerves are located in their arms. This, combined with the fact that each arm has hundreds of sucker pads capable of taste and smell, as well as their different "wiring" (octopus react to stimuli via behavioral patterns while we react via mechanical patterns), we have a creature which works normally as a single organism, with 8 arms that are basically capable of self-decision (severed tentacles have been shown to capture food and try to take it to where the mouth was). By this arrangement, we'd likely see a scenario in which the person doesn't need as big of a skull, but apparently will be capable of functioning essentially like a normal human, except this person might normally move around a bit more, letting their arms feel around for food and then shoving it into their mouths while they're talking to you as if nothing was happening, or reach their limbs out to feel your face should their owner not issue a command to stop and stand still.
Regarding faster reflexes, you'd e more or less correct, as the person's limbs will act in a way similar to involuntary reflexes, taking decision by themselves. you'll likely pull your arm faster from something hot, and you'll be able to grab things faster, but I other than that, I see no major differences. Your brain will take the same time to process images or sounds, and you won't run any faster than you already do.
Now the arthropod nervous system: in addition to a main brain, arthropods have several ganglia, dividing the nervous functions among these. This is basically why many insects can survive long periods without a head, as they basically have several secondary brains, which were responsible for keeping the body running, still intact. Crudely saying it would work exactly the same for us, in this arrangement, I don't see an increase in reactive response, as you "basically" just took certain bits of your normal brain and relocated them to your spine, but, with proper medical assistance (gotta stop the bleeding), you might be able to outlast your own head.
So essentially I'd say you'd likely be more interested in the octopus version of a less centralized nervous system. Your limbs might react faster to stimuli and things like repeated stepping or moving your hands around will likely be much more common and even seen as normal, but otherwise I'd say you won't be that different from a normal person
Octopuses as they are can solve puzzles, use tools, are naturally curious and ingenious, get bored easily, seem to be able to hold grudges, and can remember people's faces and taste (I know what I typed), so maybe your sentient creature might be more curious and exploring, as well as getting bored more easily, but that might be more from your personal decision than a trait inherent to the way their nerves are distributed throughout their bodies.