I worked at an English academy in South Korea and these books (and lots of repetition) were the primary way that we taught our young students phonics. The idea of using actions as a kinesthetic cue to remember both the name and the sound of a letter was highly effective. In our academy, we would have the students practice by saying, "My name is 'B', my sound is 'buh, buh, buh' while making the action indicated in these books.
If you were to do this, you would want to buy a set of flashcards with the letters for practice because the actual mileage that you get out of the Jolly Phonics books is not much.
This worked well for my class in South Korea but I am thinking of switching to using sign language to add a kinesthetic cue to learning letters. Why? Jolly Phonics is basically its own sign language, so why not use the real thing and give your kids some resources that they might be able to use later in life? (I realized this after I memorized actions for 26 letters and yet could only communicate with other people who use Jolly Phonics.)
Also, one final note, as an American, I did modify some of the later sounds. For example, the Jolly Phonics books have you say, "A, R, together - 'Ahhh' (with your mouth open like you're going to the dentist)." That is not how we pronounce it in America so instead we had the kids say "Arrrr" and make a hook with one finger like they were a pirate.
An interesting idea, and very effective with lots of repetition, but ultimately I think I'd go with sign language.