There's a little bit to unpack in this question. Latin was used in the Mass, the hours, and other canonical ceremonies (as well as theological discussions among learned clergy--particularly people like Alcuin who were writing back and forth to the Carolingian court), but that was only one kind of religious activity. Others were--and always have been--explicitly directed at the people. We have lots and lot of homilies in the vernacular (Wulfstan's, the Vercelli Book, etc.), visionary accounts that are kind of didactic poems (The Dream of the Rood), saints' lives (both Vitae of local saint-cults and long-poems like the Old English Saint Christopher which is a trip). To your immediate question, we even have a translation of the Lord's Prayer in old English.

Now, was translation of the Bible prohibited? Not really. In order to convert people to Christianity, you have to be able to tell them about Jesus. There is an Old Saxon (not *fully* mutually intelligible with Old English) poetic translation of the Diatesseron (a harmonized version of the Synoptic Gospels) called the Heliand. That's is an exceptional document in a lot of ways, but it's the sort of Biblical material that you would see translated in a context like 8/9/10th century England.

In Old English proper, we have:

- Two creation stories from Genesis (though one of them gets into extrabiblibal Angel stuff)
- A longish poem that covers the narrative parts of Exodus
- Some psalms
- A long poem adapting the book of Judith (which is admittedly not the most sacred part of the Bible, but it's pretty great)
- The opening of the book of Daniel
- Some direct but fragmentary quotations in law codes, homilies, etc.

How much of this is a typical peasant going to see? That's a hard question, but they're certainly going to encounter this stuff when they go listen to a preacher. Preachers whose names we know--superstar, educated, elite preachers, in other words--do use translated quotations liberally. In anonymous homilies like those in the Vercelli book--which are still exceptional and in that they were preserved--the quotations are a bit looser and shorter and they tend to harp on the big verses. It would be irresponsible for me to speculate about how that directly connects to the non-elite experience of piety! (Also, note that all of these people still interwove Biblical Theology with later tradition and authorial license. There's a lot written about the apocalypse and the last judgment that isn't super in the Bible, because it makes for a good story. Y'know, like Left Behind.)

Sources

The source texts are interesting and relatively easy to piece together for a modern English speaker. The last edition I had a copy of, a lot of them were included as practice texts in Baker's Introduction to Old English. There are some widgets that will help the untrained reader make sense of them here.

On the Bible in the vernacular, David Daniell The Bible in English (Yale 2003) has a chapter on Old English (though not a long one, because, well, we don't have a ton of surviving material to talk about, and that's not really what he's after). That book's a bit of a doorstop.