Translation:I am not going home. I am already at home.
Try, "I go not home; I am already home." if you're looking for a transliteration. I haven't tried this sentence per se; but, I've often found these present tense variations work well.... like instead of for Ich esse and typing I am eating, I type "I eat", because the perfect tense verb form always focuses on immediacy of action: and then focuses on the relative location in time or space of the other parts of the concept.
In these cases, nach seems to always refer to the the action towards; but not including the location ultimately attained. Nach can also mean the word "after" where it is more like "chasing after" but not catching; or, it can refer to something leading away from the action, like happening after.
So, for the concept of immediate, nach is anything or anywhere but the exact immediate place mentioned. It's up until you get there, or unless you've been and left there. nach Hause is just something one gets used to when one means to go home, or on your way home.
We run into similar issues with immediacy with things like vacation. "Ich habe Urlaub" meaning in German habe means one is ON vacation, as opposed to the English meaning having vacation, that is yet to be used. Haben used here, is very immediate in action, it's happening now. This shows up a lot in various forms. German focuses on things being relative to that immediacy, and it's the "little words" that do most if not all the work. Sometimes they're so little, they get combined. ex: zu + dem, two little words zum now even smaller! :-)
Here I'm expanding the term "transliteration" which means a letter by letter conversion into an alphabet for learning the sound of the word. Here I mean a word by word conversion which emulates the gestalt thinking in another language. I embed the concept of immediacy by using short versions of the English meaning. Duo almost 100% always accepts these. When you use German verbs, think of their immediacy conceptually. Ex: Ich esse (not I am eating, but...) "I eat" like NOW, "Ich habe drei wochen Urlaub" (not, I have three weeks vacation, but...) I have (right now, and so much now, I'm on vacation and enjoying it now for three weeks) lol... Yes, whatever works.
When I do this and discover something that doesn't quite work, it forces me to dwell on the concept of immediacy, and then I recognize the helper words for what they're actually doing, and their relationship to one and the other words. Positions in time and space, importance as an action, or simply knowledge as data. I'm always amazed at the complexity and detail that's included.
Read the reply I gave a few paragraphs up. I may not be the person to help because I think differently. I'm native English American (meine Muttersprache), and prefer to think and make my English sentence conform to German thinking as much as I can. They do not always make good English sentences, but the structure gives me insight into German thinking. They are not always accepted by Duo; but, are accepted more often than one might think.
I briefly mentioned above: when talking about nach and zu adding Hause, nach is closer in concept to the word after. (When going nach hause it like heading towards home, like you're chasing after getting there; but not there yet. zu hause here the word zu is more like everywhere nach is not. It zu includes actually being at that place, and not just "to" the place.
The reason has to do with German likes to express something's relationship to another, a lot! Whether it's relationship to time, or location, or yourself, or someone else. German has a lot of words to put things in their place. A lot like English; but, not always the same way we do in English. This is what makes it harder for us. We're not as strict in English and sometimes kind of lazy about things and where and when exactly, and only add details if someone asks.
Most often "nach" means after, and if you find it being used in a different way, try to find a way to use the word after, to describe the relationship. Like I did here: Nach Hause, would be after House but how. By heading home, one is going toward, chasing after, wanting to to be home. Ich fahre nach Berlin I am driving toward Berlin, or I am driving after Berlin, to get there. After a while, it will come naturally. Ich fahre nach Hause. So picky about fine details. Zu, towards including actually being there. Nach, towards, but not there yet. Ich bin zu Hause. The verb "be" or bin here is so immediately now, it's not possible to be heading to the house. Here 'zu' means not to, but there, or at. Like beim Haus. Zu Hause means at home.
These tiny words have so many ways they can be used, mostly it's just finding a way to get them to stick, conceptually, with the sentences and words used with them all together in on place in the mind. It takes practice and time, there's no easy way out. The same for people who learn English. They're like idioms, we just have to think about them a lot, and use them.
Ich gehe nicht nach Hause. Ich bin schon zu Hause.
I go not chasing after home. I am already to (no..) AT home. I go not after home; I am already to home. (It's the concepts that form, when you think about the German words, when you read it. It's the pictures you make in your head while you read it.
Ich gehe nicht nach Hause. Ich bin schon zu Hause.