18th century Russian ikon of the Biblical prophet Nathan (Natan)
The English and French name Nathan comes from the Hebrew Natan (he gave). Many people are familiar with the above-pictured Prophet Nathan, who served under King David and took him to task for cuckolding Uriah and sending him to die in battle.
Though it’s long been common in the Jewish world, this name didn’t become popular in the Christian world till the Protestant Reformation. While we think of many Biblical names as going either way today, they were once considered exclusively Jewish.
Nathan entered the U.S. Top 100 in 1972, at #79, and attained its highest rank of #20 in 2004 and 2005. In 2019, it was #55. Nathan is also popular in France (#18), Belgium (#14), Switzerland (#41), Scotland (#45), Italy (#50), Ireland (#61), New Zealand (#70), The Netherlands (#77), Northern Ireland (#83), and England and Wales (#104).
Israeli human rights activist, politician, and author Natan Sharansky (né Anatoliy Borisovich Shcharanskiy), centre, born 1948
Other forms of the name include:
1. Natan is modern Russian, Georgian, Polish, Galician, Serbian, Ukrainian, Dutch, Czech, Slovak, Scandinavian, French, Finnish, Icelandic, and Croatian. Alternate forms are Natán (Spanish), Nátan (Faroese), and Nátán (Hungarian).
2. Nafan is the traditional Russian form. I’m not exactly a big fan of Russian names where F takes the place of TH in the middle of the name!
3. Noson, or Nosson, is Yiddish. I’m also not a fan of Yiddish words and names where T is pronounced like S. Nails on a chalkboard 99% of the time! There’s a reason modern Hebrew uses Sephardic pronunciation instead of Ashkenazic.
4. Nâtat is Greenlandic.
5. Nâta is also Greenlandic.
6. Neihana is Maori.
7. Natuš is archaic Sorbian.
Nathaniel Bowditch (1773–1838), American mathematician and father of modern maritime navigation, painted by Charles Osgood
The English name Nathaniel comes from the Hebrew Netanel (God has given). The variation Nathaniël is Dutch. Like Nathan, it also was largely found in the Jewish community until the Protestant Reformation, when many Biblical names were suddenly proudly embraced by the Christian world.
Nathaniel was in the U.S. Top 100 from 1978–2015, with its highest rank of #60 in 1998.
Other forms of this name include:
1. Nathanael is an English variation. The form Nathanaël is French and Dutch.
2. Nataniel is a rare Spanish and Portuguese form.
3. Natanael is the more common Portuguese and Spanish form.
4. Natanail is Macedonian and Bulgarian.
5. Natanaele is Italian.
6. Natanayil is Quechan, an indigenous language spoken in the Andes Mountains in South America.
7. Nathanail is modern Greek.
8. Nafanail is Russian. Again, it’s nails on a chalkboard to see and hear an F in place of a TH in the middle of a name!
Female forms of both:
1. Nathana is English.
2. Natana is Hebrew.
3. Natanya, or Netanya, is Hebrew.
4. Nathanya is a rare English form.
5. Nathanielle is English and French.
6. Nathaniella is English.
7. Nathaniela is English.
8. Nathaniele is English. The variant Nathaniëla is Dutch.
9. Nathanaelle is English.
Onomastics Outside the Box 
I don’t really like Nathan, or even more so the nickname Nate, but I do like Nathaniel and the nickname Nat. I like Nathanael even more so.
Here in Poland the letter N is quite trendy for babies now, and Natan was #42 last year. I don’t know either any Polish Natans or Natanaels/Nataniels but just as even five years ago pretty much the only place where I’d come across these names would be the Bible, now I do see both Natan and Nataniel not super often but regularly online as names of little boys. Nataniel is #341. Sadly Natanael is all the way down at #454 which sort of makes sense because “ae” is not a popular sound combination in the Polish language but then the foreign Nathan and Nathaniel are higher in the ranking. Because I personally prefer Natanael (Nataniel feels much more softy) I am inclined to think it kinda sucks, but oh well. 😀
Nafan and Nos(s)on do sound really unpleasant to me and I mostly agree with you about the f or s instead of th thing.