Hi Josh. Full disclosure - I’m not a scientist, let alone an astrophysicist. So while I feel I may be able to answer your question, maybe someone can butt in and correct me if I’ve made errors.
The cosmological redshift of electromagnetic waves is due to the apparent movement of the source traveling away from us, similar to the doppler effect. I say ‘apparent’ because, when talking about the expansion of the universe, it’s worth noting that the objects aren’t actually moving from their coordinate, but rather the expansion of space everywhere increases the distance between objects, giving the appearance of everything moving apart. This increased distance between things means that light takes longer to travel between them.
As far as I know, the extra time it takes for light to travel due to expansion is not a relativistic effect, such as time dilation (wasn’t sure if this is what you meant when you said “time shifted”). It is merely a matter of greater distance = greater time to traverse it. As an example, a signal sent to us from an object that is initially 1,000 light years away will take 1,000 years + a bit more due to the extra distance from expansion by the time is reaches us. So it is not that we are perceiving radio signals from the New Horizons probe as taking longer from our reference frame, because the expansion/increasing distance is also seen from the perspective of the New Horizons probe.
I’m not sure, as I don’t know the math (there are equations to calculate expansion, as the rate of expansion increases for objects the further away they are), but the expansion between us and the New Horizons probe would probably be extremely small, and perhaps negligible for practical reasons.
A cool tidbit is that, from the perspective of the light (radio signal), it would not experience any difference from expansion as light does not experience time. From the perspective of light, everything is contracted so that there is no distance at all!