Do it right as you deliver the final product. That is the apex of their happiness with you.

Include it as the conclusion of your meeting with them to discuss the final product. Make sure they know this is coming and are set up for it. Provide a highlight script for them to follow. You don't want word-for-word, but you do want sample sentences around which they plug in their own experience.

Eg. "We were in Tough Situation #1 when we found Supplier J. One of the most amazing things about working with Supplier J was {fill in the blank}. What we love most about our video is {fill in the blank}. Now that we have the help from Supplier J, we've moved out of Tough Situation #1 and are able to {fill in the blank}."

This qualifies viewers of the testimonial who are in the same bad situation as your client was.

Get this done by video: you're the pros at this so you should be able to figure it out, and make sure it is completed right then. Not later. Don't let them off the hook. Set it up as the expectation of the conclusion of the process and have them ready to do it.

As soon as you let them get away without providing the testimonial at this point, as you've noticed they don't need you anymore. So while they may be well-meaning and want to help you, you just aren't that great a priority anymore. They have other fires to fight.

If you miss the opportunity, you're probably going to have to orbit until they return for a new video. Then get the testimonial before you get to work on that new product.

Getting testimonials is a HABIT, like walking for 20 minutes after work. People say, "Oh, I'm too busy" and the like but that is absolute nonsense. Once you start making time for this important marketing tool, you'll find it is a natural requirement and you can't proceed without getting it.