"That child does not go home."
Translation:वह बच्चा घर नहीं जाता।
Only objects of postpositions take the oblique case. In this case, there is an implicit postposition before जाता and its object is घर. Therefore, घर is in the oblique case but its form in the oblique case is also घर.
बच्चा and जाता are not in the oblique case.
If you're having trouble identifying the objects of a postposition, consider the sentence with the implicit postposition written out- वह बच्चा घर (को) नहीं जाता - 'That child does not go (to) home'.
The object of को is the part of the sentence that answers किस को?- Where to?
Therefore, घर is the oblique case.
The sentence you wrote - वे बच्चे घर नहीं जाते means 'Those children do not go home'.
Neither. It means that the child (usually/regularly) does not go home. Eg: 'The child doesn't go home after school on Wednesdays. He attends after-school sports' - 'वह बच्चा बुधवार को स्कूल छूटने पर घर नहीं जाता। वह स्कूल के बाद वाले खेल-कूद में शामिल होता है। '
I agree that the sentence (both in English and in Hindi) does not make a whole lot of sense without additional context.
"That child never goes home" would be 'वह बच्चा कभी घर नहीं जाता'. कभी here works like 'ever' so the sentence is literally 'The child doesn't ever go home'.
"That child will not go home" is 'वह बच्चा घर नहीं जाएगा'. जाएगा is the future tense form of the verb.