You probably don't need to worry about the wood itself. The biggest problem is if the branch was sprayed with insecticide or dormant oil, and usually this happens in the spring before the trees bud out.
You will find many sites that list peach and plum as dangerous woods. I looked at many of those and didn't see any scientific references as to the wood being toxic. The seeds, or pits, in the fruits of these trees do contain compounds that are mildly toxic, in particular weak cyanide compounds. Apple seeds also contain these compounds, so perhaps people generalize that if the seed contains a poison,mother wood must also - but this isn't always true. For example, mango and cashew are related to poison ivy and the sap of both contains urushiol. You should always wash mangos carefully, because if you come in contact with the sap on the skin you can develop an allergy to mango. Eating the fruit that encases the cashew nut will give you a terrible allergic reaction, but the nut itself is tasty and safe.
On veterinary websites I found lists that show peach and plum as safe. A very reputable site,
Non-Toxic and Toxic Woods | Center for Avian Rehab, lists both these woods as safe. You will find sites that come down on both sides of the argument, but I did not find one site that documented a single death of a parrot from exposure to those woods, or any documentation of birds made sick. The most I found was that the wood may contain toxins and should be avoided.
I'm not a vet or avian researcher so I can't give you an expert opinion. My educated opinion is that the greatest danger is from any pesticides or natural fungi or bacteria or yeasts that might be present on outdoor branches. I'd keep an eye on birdie, make sure birdie is eating well and acting healthy. You can call an avian vet to see if they think the bird should come in, and an avian vet should be able to give you good advice. I don't believe you need to panic or worry if it's just one chewing session, but any wild, undisinfected branches can be dangerous so no more free range chewing!
