Think about it:
Husband/Wife has “spouse.”
Mother/Father has “parent.”
Daughter/Son has “child” or “progeny.”
Brother/Sister has “sibling.”
Yet there is no gender netural term for uncle or aunt. And no gender specific term for cousin.
Why not?
Actually, there are gender-neutral terms for uncle and aunt, and gender-specific terms for cousins. But not in English.
Indeed, different languages can have very different ways of distinguishing relatives. For example, I believe Cantonese has different names for uncles on the mother’s side vs. the father’s side.
And in Japanese, sibling words are not only divided by gender, but by relative age as well. One can’t easily talk about their sister in Japanese without specifying whether she is a big sister (ane), or a little sister (imouto). My understanding is that the “big” sibiling words are used in cases where there is no easy way to tell, or when using them metaphorically, so nuns for instance I believe would always be “ane”.
Some languages are very strict in having both feminine and masculine words for just about everything. This is more typical in the west than the east. I think the idea of gender neutrals for everything are a very recent idea in English and lead to strange things like “Womankind” “he/ir” etc that dont catch on.
If a word for “uncle or aunt” were needed, it would be coined. Similarly, if a word that differentiated the sex of cousins were needed, it would be coined, too.
For instance, the English language got along just fine without a word meaning “brother or sister” until around 1900, when “sibling” was coined.
Why aren’t there? Because not enough people have felt the need for such words. Make up some words with these meanings if you like. Sprinkle them into your conversation, your message board postings, and anything else you write. (Try to make sure that, in most cases, you either explain what the word means, or that the meaning is reasonably apparent from the context.) Maybe the words will catch on and become part of the language.
More probably, however, they won’t, because, except perhaps for professional genealogists and anthropologists who study kinship systems, a real need for making such distinctions or non-distinctions rarely arises. Actually, I would not be surprised if anthropologists and/or genealogists actually do have words like this, but they have remained as obscure technical jargon because nobody else has much need for them.
Was there ever such a word that just fell out of use?
And there’s always “parent’s sibling”. Not one word, but it works. Start saying it and see if it catches on. 
In Danish there are gender specific words for cousin: fætter (male) / kusine (female). In Norwegian there is only the word “fætter”, in Swedish only the word “kusin”. My guess is there originally was a Latin word (cousin) and a Germanic word (fætter), and most languages chose the one or the other. Danish chose the both.
English has also only the same word for grandfather/grandmother on the mother’s and father’s side.
Yep, and different names for aunts and grandparents on the mother’s or father’s sides. Also distinction between older and younger brothers and sisters.
And also by whether I’m talking about your big sister or my big sister.
I’m always frustrated by the lack of a gender-neutral term for niece/nephew. With the boyfriend’s sister expecting, I’m sick of saying “Your niece-or-nephew”. I coined “Niecephew” (though I’m certain I’m not the first to do so) but it just isn’t catching on.
Not sure what you mean by “originally”. Cousin derives from Latin, but it probably entered those languages as it did in English-- through French.
I use “sobrinos” (Spanish) to refer to my brood of nieces and nephews. Maybe it will catch on.
ETA: Maybe someone who actually speaks Spanish will help determine if this helps in your case. My understanding is that “sobrino” is “nephew”, “sobrina” in “niece”, and, as tends to be the case with Spanish, you use the masculine ending for a mixed group. Not sure if the same holds true for one of yet-determined sex.
Nah. The word you want is nephling.
Every once in a while I run into someone who tries to convince me that English is an inferior language to others, particularly other Romance languages, because it is less “specific” of a language. When I ask them for an example, they invariably respond, “Well, in English there aren’t separate words for a ‘female cousin’ and a ‘male cousin’.” My response is always, “So?”
Carry on…
There is no word “fætter” in Norwegian, but there is a word fetter 
And it specifically means “male cousin”. “Female cousin” is kusine.
Male and female cousins together can be called søskenbarn, literally “sibling-children”, which sounds like it should mean nieces and nephews together but for some reason doesn’t. Some women refer to their nieces and nephews together as their tantebarn, “aunt-children”. I’ve never heard a man talk about his onkelbarn, however.
I believe the logic is that a pair of sibling-children are the children of a pair of siblings.
In Swedish, this usage of “syskonbarn” has become quite uncommon (due to “kusin” filling that lingual slot) and these days the term almost invariably is used to mean niece/nephew.
Your understanding is correct. Of course, there happens to be this branch of Stupid Feminists who have decided that using the masculine as the neuter is Inacceptable, and who would write either sobrinos/as or sobrin@s, pronouncing either sobrinos-sobrinas, sobrinosas, or sobrinoas. They’re the same people who get hysterical (1) when someone refers to the gender-still-unknown future-baby as “el niño,” “el muete,” “el mocete,” “el bebé,” “el crío” or “la criatura.” I’ve even seen one start climbing the walls over “lo que viene” (lit. “whatever’s coming”).
Every family has an idiot cousin (which may be of either gender).
(1) word chosen with utmost care.
-Yoink-
Thanks!