Wooo! I play both of these characters a lot and I have played the match-up a few times so I know a thing or two about this.
Generally as Zelda you want your opponent to approach you and Peach is no exception. Fighting a Peach that approaches you is perhaps even more beneficial in this match-up than versus other characters since Peach's mobility is sloooow, making it easier to punish stuff like floating approaches, grounded approaches and her landings. Nair works great vs a floating Peach and vs a grounded Peach things like Fsmash, Dtilt, Dsmash, Jab and Nayru's are very effective. Watch out for the turnip, you can reflect it well vs grounded Peaches. Try to avoid the turnips though, run away and let her Peach chase because you really don't want Peach to zone you and lock you down. On shield Peach is pretty solid though, a lot of her stuff is safe so don't rely on too much on shield or whiff punishes. Let her chase you and punish her mobility options, jumps and landings.
However, if the Peach player knows the match-up and doesn't float approach 24/7 or approach in general... oh god that's where the horror starts. Turnips are a pain. Her plucks are pretty laggy though so try to stop her from plucking one and avoid getting hit or even shielding one, because she can start her pressure really well from a turnip throw. Peach also wins the spacing game easily because she moves back after hitting on shield so almost everything she does is almost unpunishable. On top of that, everything she does is much faster than anything Zelda can do. Zelda will have to be really patient and punish stuff like her jumps and landings with moves like Fsmash, Ftilt or Phantom. Don't play her spacing game because you will lose, have her chase you all the time. Zelda's Usmash nerf really hurts in this match-up since stuff like Peach's Dair has become even more of a pain...
Be patient when you get hit, Peach is good at pressuring. Occasionally throw out a lightning kick while landing. It's unsafe but it throws Peach out of her rhythm and sometimes you might even hit the sweetspot. Most Peaches will respect your landing more after getting killed by kicks. Escaping with Farore's works fairly well at times since Peach's dash is slow but Farore's also has looong ending lag. Use it if you know you'll land safely or far enough for Peach not reaching you in time. You really don't want Peach to land another hit on you since she can continue her pressure for eternity, and she also kills much earlier in this game.
tl;dr let her chase you and be patient. This match-up is a true test of patience. I think it's pretty piss for Zelda though... It isn't very hard to hit Zelda with stuff like Dair so Peach can start her pressure easily and out spacing Zelda isn't hard either with her float since Zelda doesn't reach her too well when she's floating (R.I.P Usmash, Zelda's Nair works decently well though) or moving back after doing an aerial. Peach gets worse when she has to chase her opponents but she doesn't need to chase Zelda since she has turnips, aerials, a float and Zelda's mobility is just as slow as Peach's is.