Allons-y. Let's go (there).. -actual imperative mood, actual 1st person plural (we).
On y va. (No hyphens). On - is a common casual substitute for the more formal nous, but only as a subject (on - has other meanings as well). Literally: We go / are going (there). indicative mood, and a statement. There is no imperative form for "on".
Practically speaking, they both can be used to mean: let's go, but "on y va" is more casual and more likely in daily spoken French.
On y va -- can also be a simple statement, rather than a suggestion. We're going.
On - can also be an indefinite subject, when the speaker is not entirely certain what the subjects are. So, on y va -- can also mean: They/someone/everyone etc are going (we don't know exactly who). But that meaning is more likely in more formal French.
It has nothing to with future or present. The meanings are the same when "on" means "we".
taught French; native English speaker.