Let me make some remarks on what you say above: "Imagine you want to say something like "from us to you [plural]" (where "from" indicates ablative and "to" dative). Since the order is usually uninformative in Latin, nobis vobis is not precise enough. Would something like a nobis vobis be enough?".
As pointed out by Joonas, context is important here. For example, if the verb dare is involved, the example from Plautus given by brianpck is indeed very natural. If the verb is transire, the most adequate construction would be another one: e.g., cf. August. Bapt. 1.10.13: qui transeunt a nobis ad vos.
Finally, please let me make a more general remark on what you say above: "the order is usually uninformative in Latin". This is often said in some (traditional) Latin grammars but it is not correct. Quite the opposite! Despite appearances, word order is usually informative in Latin. For those of you who are interested in word order in Latin, three important recent works are:
Devine, Andrew M. & Laurence D. Stephens (2006). Latin Word Order. Structured Meaning and Information. New York: Oxford University Press.
Devine, Andrew M. & Laurence D. Stephens (2019). Pragmatics for Latin. From Syntax to Information Structure. New York: Oxford University Press.
Spevak, Olga (2010). Constituent Order in Classical Latin Prose. Amsterdam/Philadelphia: John Benjamins.