I saw a couple other threads with slightly similar questions. But I want to stress with this one, that the focus is what instrument and instrument family would be the easiest to build anywhere in the world if we started from the beginning of human civilization again. Also, definite pitch capable designs only! Even if it’s only capable of one pitch at a time.
End-blown flutes are very easy to construct from readily available materials.
You only need a reed or other hollow-stemmed woody plant, with some way of cutting and shaping it (stone tools will suffice.) You can either make a set of pan pipes, with the different lengths forming different pitches, or make finger holes to create something like the North African/Middle Eastern Ney.
Transverse flutes and fipple flutes are also easy to design, and were made (for example from bones of large birds) in prehistoric times.
Flutes and Drums
Flutes are the oldest known instruments in the archaeological record. Some date back up to around 50,000 years old! Also, it's argued that Neanderthals made https://en.wikipedia.org/wiki/Paleolithic_flute It's also thought that a prehistoric flute can be attributed to neanderthals though this is debated. https://www.nms.si/en/collections/highlights/343-Neanderthal-flute
No pitch but drums have probably been around for similar periods of time, in the form of skins stretched over stuff and banged, but these soft materials don't preserve through time in the archaeological record so we can only speculate.
or Wood Chimes
They can be made with a number of naturally grown hollow plants. Materials available all over the world, require minimal tools to construct, and require no complex assembly. You don't even need to space or cut complex holes for multiple notes, just trim to length to tune.
Literally Everything Less Electric
Pitched Percussion. This is simple! If your world is Post-Pockyclyptic, then you've got loads of already existing metal, stone and wood bars around. You just need to harvest, shape to the desired form and carve down or add mass to tune. Add some carved sticks and connecting strings and you've got a carillon.
Even if your setting is naked humans dropped off on a new planet, it only takes very basic skills to shape bone, wood, stone and leather into fully functional and well tuned xo-phones.
A hollowed log, a piece of prepared skin and some ropes are enough to make a tunable drum! Metalwork can get you nakers and timpani.
Bamboo, long hollow bird bones, clay, hollowed wood all make the basis for wind instruments. With very basic metallurgy you can fashion simple system flutes, clarinets, oboes and cornetts. The better your woodworking skills, the bigger the instruments you can make.
But really, anyone can make a well tuned whistle or flute or shawm from a piece of pipe and a reed.
Wood and horn and some basic techniques can yield you a pibgorn, a cornett, an oliphant; add some leather for bagpipes and the like.
A smallish log can give you the basis for a dulcimer, a harp, or a zither. You've got many string choices from gut to silk to other plant fibres. Add a little metallurgy for brass strings! As your woodworking improves, you can thin the walls of the instrument body and make any stringed instrument that exists now.
Even keyboard instruments are not beyond reach! You won't be able to make a modern piano until you get cast iron and steel, but harpsichords, clavichords, regals, organs, harmoniums, wheelharps are all within reach once you've got carpentry skills and some decent tools.
For a percussion instrument with definite pitches, try a xylophone.
The difficult part is simply to carve a suitable piece of wood so that it makes the desired note when struck. Once you have a set of these pieces of wood, you can place them on straw to create a strohfiedel.
See the Wikipedia article on the Chelys. This was an early form of lyre. A myth (there are many variations) had it that Hermes found the dried corpse of a tortoise that had dried in the sun, leaving seven fibres stretched across the shell. That does not explain how the lyre got its extended neck, but once you have the basic idea, the rest follows.
Another possibility if resetting the world left lots of scrap metal is the Mbira or Thumb Piano. Africa had these for thousands of years. They can use wood instead of metal.
One more instrumen I can think of is the Kalimba or its close relative the Mbira.
A series of metallic spikes of different lengths generate different pitches when plucked or striken. A wooden box would serve as a resonator and metallic spikes are scraped from older artefacts. How far the civilization is "reset" can determine how adept this civilization would be in processing the metal ores and from what sources. On the other hand, wooden spikes, presumably bamboo, have been used for the task as well. See the history of the tool here [source]
I understand that stringed instruments have already been suggested, but this answer aims to specify a motivation for them, though speculatively at best.
If the world reset, I'd wager that hunting would be important again and I would assume that bow and arrow designs would be developed rather quickly.
Of course this depends on how much history and folklore is available. But it's easy to see that different sized bows, custom made for different hunters or for different purposes, would make different sounds when the strings are plucked.
Assuming further that a good hunt, and given that cooperative hunting is much simpler and lower effort than farming for small, clan like pockets of humanity, leads to leisure time ( personally I think our common intuitions about modern society and the role of farming are short sighted ). I'd also guess that variations on bows to make different sounds would be considered.
Animal calls and bird songs are natural and serve to inspire curiosity and creativity in humans. So it's not a big stretch to imagine stringed instrument abstractions, along with the sounds that we hear in sea shells, and so on.
All simply ripe for the fertile mind to play with.
Sure, pipes, flutes, ocarina, drums, etc. are going to be pretty easy to makes once you get past your basic survival needs, but they still require time and tools to make, and in those first few years when every hour spent crafting must be spent in the name of survival needs, setting aside the time to make even a simple flute will be a luxury not worth taking. However, there is one common tribal instrument that takes no effort at all to make: clap sticks.
All you need to do is pick up two sticks and knock them together. At most, you will need to spend 5 minutes scraping off any excess shoots and bark. When you clap them closer to your hands, they make a higher pitch sound, and when you clap them closer to the ends they make a lower pitch sound; so, they are not specific pitched instruments, but the pitch a good clap stick makes is definite.
Also, unlike most other instruments, you don't need to finish it to use it. Over time, you can refine your clap sticks in small incremental steps. You may at first start off just taking any sticks from your nightly firewood pile. Over time the sticks that make more definitely pitched sounds get kept and the less-definitely pitched sticks are returned to the your firewood pile. So before long, you will have some nice, straight, dry, knotless hardwood sticks pulled aside for your nightly jam sessions without having to invest any real work into making your instrument.
Clarifying Definite Pitch
Since this is not a common term outside of music theory, I believe there is some confusion about what this means. Definite pitch means that a sound follows a wave pattern that is all more or less one pitch like a key on a xylophone. An indefinite pitch is like the sound of a tambourine or rattle where the sound it makes is a wide range of simultaneous pitches.
Tuned/tunable instruments are called specific pitch instruments, but there are also a wide range of unspecific, definite pitched instruments which includes most slapsticks, hand drums, etc.
Instruments that are not both specific and definite pitch are often called unpitched instruments. But Google does not seem to know the difference; so, when you look up if certain instruments are indefinite pitch, you will often get answers for unpitched instruments instead; so, if you Google clap sticks, the results you get can be very misleading.
This said, slapsticks can be either pitched or unpitched based on thier design and use. If they are straight and made out of an aged homogenous wood like oak, they will be pitched. If they are bent, tapered, greenwood, or made out of a stick with distinct heartwood and sapwood like hickory, then it will make an less pitched sound. The technique you use can also make a difference between if you get a definite or indefinite pitched sound out of them much like how a drummer can get either a definite or indefinite pitched sound out of a djembe, though generally speaking, the term definite pitched instrument just means that it CAN make a pitched sound, not that it always will.