When did British and American crochet terms diverge?

In crochet basic stitches are called different things. For example a single crochet in America is called a double crochet in the UK, a double crochet in America is called a treble crochet in the UK, etc. I think (and this is a guess) that they are both consistent, with the American versions named after the height of the stitch and the UK version named after the number of loops on the crochet hook.

When did they diverge?

Crochet feels like an old craft that must have been commonplace around the time that Europeans settled America. Describing crochet stitches must have been similarly commonplace so what would cause these two usages to develop - is there a history of crochet terms that would chart this out?

You're making an incorrect assumption about how old the craft is. From Wikipedia:

Lis Paludan theorizes that crochet evolved from traditional practices in Arabia, South America, or China, but there is no decisive evidence of the craft being performed before its popularity in Europe during the 19th century.

So the terms are presumably different for the same reason that the terms used in the US and the UK for parts of automobiles (bonnet, hood, boot, trunk) are different: the terms were settled on independently on different sides of the Atlantic.

Crocheting For Dummies (2010) by Susan Brittain, Karen Manthey and Julie Holetz says:

Just as hook sizes differ depending on where you live (as explained in Chapter 2), the crochet terminology used in the United States is different than that use in the United Kingdom and Australia. For example, what's called a single crochet in the United States is called a double crochet in the United Kingdom. No one is entirely sure how this different crochet terminology came about, but one theory is that the UK terms refer to the number of loops you have on the hook after you draw a loop through the next stitch and that the U.S. terms refer to the number of times you move the hook to complete the stitch. For example, after you've drawn the yarn through the stitch in the previous row, you have two loops on your hook or a double loop, hence the United Kingdom's double crochet. To complete the stitch, you only have to draw the yarn through both loops on the hook once, which is a single movement, hence the United States' single crochet.