Well I'm not an English native, but that sounds to me like a weird word order. I'd say either "He is going home quickly", or more probably "He is quickly going home".
Can it not take on the meaning "He is going home early" ? Or if you use "szybko wraca do domu" could it take on such a nuance?
No. Firstly, it doesn't seem like a correct word order, but even if it was, "idzie" needs Present Continous, not Present Simple.
Correct English grammar puts the adverb next to the verb it modifies; before or after probably depends on context. Here, quickly goes home is likely best. Goes home quickly is common but not technically proper. Goes quickly home sounds odd to me, like you're trying too hard to follow the rule, but may be less odd outside of America or even in a southern or coastal dialect.
Well, but that's more of an interpretation than a translation. You didn't use any word for "szybko" nor for "idzie". Your sentence translates to "On śpieszy się do domu".