It probably wouldn't help, and TAOCP from the beginning is too many pages. It would help to read the division algorithm chapter.
I introduce my own notation from medium->small division onwards. If it's any help u'', v'' are the limbs that the division instruction sees (the ones pertinent to qhat) and u',v' are all the lower limbs.
I appreciate the info but it’s unfortunately no help.
I have a passing interest in mathematics and especially comp sci but I routinely find myself stymied just by trying to understand the notation.