Are there any English Bible translations whose primary text for the OT is the Septuagint? The translators of most English Bible translations used the Hebrew Masoretic Text as their primary source for the Old Testament, though they will use the Septuagint and other texts to critically analyse the Hebrew text. (You can see evidence of this in footnotes that note that the Hebrew says one thing and the Greek says another.)

Are there any English Bible translations of both the OT and the NT whose primary text for the OT is the Septuagint?
There are a few. Some of them are published as academic works, while others are intended for the common use and worship (especially in Eastern Orthodox Churches).

The first was The Holy Bible containing the Old and New Covenant, translated by Charles Thomson in 1808 (though he did not include the apocrypha). It can still be bought today. The translation was revised and enlarged by C. A. Muses in 1954 as The Septuagint Bible.

Sir Lancelot Brenton published one in 1851. It has continually been in print since then and was updated as LXX2012: Septuagint in English 2012.

There is also the New English Translation of the Septuagint (NETS). Even here, they have to do textual criticism as the extant copies of the LXX don't all agree. In the translation principles, the committee states they have a target audience "that closely approximates that of the NRSV."

The Eastern Orthodox churches in English-speaking countries also have several options available for English translations based off the Septuagint. In 2008, The Orthodox Study Bible was released. It uses the semi-critical Septuaginta of Alfred Rahlfs as its base. Peter King, SJ, has also released a translation of the Septuagint in four volumes: The Pentateuch 2010, The Historical Books 2012, The Wisdom Literature 2008, and The Prophets 2013.