I don't think anyone who read few fantasy stories that predated Harry Potter would ever have this kind of consideration. This is because wands were a widespread idea. Moreover it did not only exist in books/stories but were also physically used to make things look more magical.
Others have already pointed out several examples of wand examples from stories. Let me just add that the use of wands in pop-culture was (according to this Wikipedia article) widespread by several Disney movies in the first half of twentieth century.
But, wands are also a popular attribute whenever magic is shown to people. Just consider the illusionists shows. It may be less popular now, but it you look at any young magician set (predating HP) it consisted of a mandatory wand (as well as some cards, usually a hat and a cloak).
Just a random example taken from simple Google search for young magician magic set:
As you may read for example on this page the usage of wands is described in old documents referring to magic, dated as early as 13-th century with objects resembling wands found dated 3-rd A.D. century or even B.C.
See just a single example picture from the aforementioned site:
Other example may be lees obvious, but it might be worth mentioning here. I don't know how local this custom is but in Poland there existed a concept of "różdżkarstwo" (dowsing) - a skill of finding underground water with a use of wands (in Polish wand is translated to "różdżka". The tools used in the process are called the same; according to Wikipedia tools used for dowsing are called in English rods or twigs, but the more general rhabdomancy uses also wands). Now take a note that those wands were different than what we usually imagine. I'm aware of two types of wands used, both require two hands.
The first type actually require two wands, L-shaped (with a very long one part and a short handle perpendicular to the rest of the wand). The dowser ("różdżkarz") holds one wand in each hand with the long part parallel to the ground and pointed to the front and starts wandering in the area where they look for water. The wands in his hands react moving to the sides, supposedly aligning with the underground water streams. Places where the wands cross are the location of streams crossing that are either useful to build a well or needs some negative field reduction tokens (like magnets). Those places also aren't suitable for sleeping (negative energy flows over those streams).
The second type of wand used in the process consists of two curved pieces joined at one end (I think one of the bones was originally used for that or some properly shaped wooden stick). The dowser hold the separated ends in each of his ends with the joining ends pointing outwards and again walks in the area. The wand moves in their hand up and down. When it points vertically down, it's again a place where underground water is (what to do with that is similar as in the previous case).
You can see examples in those Youtube videos (sorry, they are in Polish only):
The way wand is used also varies significantly depending on the story. Let me limit myself to modern stories and other productions (OK, Lord of the Rings is already about a century old, but still...) and use just few different examples.
In Lord of the Rings wand (technically it's a staff but works the same) is necessary to do (powerful) magic. When Gandalf enters the court of Edoras (in Two Towers) he tricks guards to let them think it is a wandering stick in order to be able to take it with him inside. He needed that to duel Saruman's charm on king Theoden. Later (in Return of the King) Gandalf breaks Saruman's staff/wand removing his ability to cast spells (he can still charm to some level with his speech though). This idea is pretty similar to one in Harry Potter (so as you can see it's Rowling who borrowed the concept), where wand is said to channel the magical energy and it is clear that it is required to cast any spell (e.g. in HP and the Order of PhoenixMs. Figg requires Harry to keep his wand to be able to react quickly if Dementors come back).
In Discworld series by Terry Pratchett wand is useful for witches to do some magic, but most of it can be done without one.
In Nethack game (dated 1987) the wand is a powerful magical object that can be used by anyone (not necessarily a wizard). each type of wand has it's own way of working and you can do only one kind of magic with the specific type of wand. It can also be used limited number of ties and can be recharged once used up (but each recharging increases risk of breaking it). Finally breaking a wand causes an uncontrolled explosion of magic (usually it turns out quite deadly for the adventurer) - contrary to LOTR where it just removes magical ability of the wand wielder or even further moved HP where a wizard or witch can use any wand, however their wand (or the one that accepted the wielder as its owner) is more efficient.
As you can often find here - it's your world and you can build the way your wands work as you please.
The only things that seem specific to HP world are:
- the construction of a wand (a core of a magical animal and a surrounding wood of specific kind implying the capabilities of a wand to some level)
- the wand-owner relationship
- the relationship between wands (wand twins made of the only two feathers of on phoenix)
- The Elder Wand as a product of Death and its capabilities
- the attributes of the wand that seem important to impact its ability (to some level, these are mentioned as core, wood, length and stiffness/flexibility). Yet I believe if you decide to make length of the wand (and some other things, other than in HP) important it will be impossible to say you borrow the concept here.
- list of spells (and for some - the effects caused by the spell if they are HP world specific)
I think it will be a good idea to read Pottermore page about wands to be aware what might cause a problem. Especially look at the details I mention above so that you are super-cautious not to copy that in your wands concept. Make sure to follow all links you will find there (every block is a link!).