Card, Moran and Newell’s Goals, Operators, Methods, and Selection (CMN-GOMS) family of models can be used to represent cognitive load. Many are familiar with the KLM-GOMS version, which models the physical steps in a task, but there are also elements and versions of GOMS (e.g., CPM-GOMS) which seek to model the cognitive steps in a task. The raw number of steps, the time to complete the cognitive steps (allowing for parallelism), and the number of levels in a task tree each represent different aspects of cognitive load.
While KLM-GOMS is relatively easily applied with little training, other flavors of GOMS take more expertise and effort to apply, which may account for them not being used as much. Instead of measuring and comparing cognitive loads, it’s easier to analyze a task to identify what cognitive load can be shifted to the computer (e.g., the current page replicates information from another page to reduce memory burden; the page computes and displays differences and percents to reduce mental arithmetic). In this way, we approach the minimal feasible cognitive effort without actually measuring how much it is.
As for the impact of cognitive load, usability seeks to minimize the time, effort, and errors in using a product. Each cognitive step you impose on a user represents a little more time, a little more effort, and a little more chance of error. Shifting these steps to the product UI improves usability as long as the product does it faster and more accurately than the user.