The most important single qualification for an experienced pilot who wishes to become a test pilot is a quality engineering degree, preferably in Aerospace Engineering, and experience in design or flight test. TPS is highly touted, but not imperative, as an engineering degree is quite a bit more valuable in terms of actual knowledge needed in the job of being a Manufacturer's test pilot. Surprisingly few graduates of military test pilot schools are engineers.
TPS is actually quite poor in preparing TP to do real development work, as the course work at all the TPS I am familiar with is almost exclusively devoted to the testing needed to assure compliance with requirements. Virtually no coursework is associated with the essence of Aero Engineering, or of structural/dynamic concepts that are usually the source of development problems and therefore the root of the manufacturer test pilot's challenges.
In industry, a sharp engineer, usually with test or design experience in the organization, is quite able to become a test pilot. This is doubly true in General Aviation.
I represent that pool of non-TPS pilot, I was Chief Project Pilot on two development aircraft, 3 research aircraft, and also Chief R&D Test pilot for a major helicopter manufacturer. Our Chief of Flight Operations and the senior test pilot on most of our product line helicopters were top-notch graduate engineers with experience in test or design. They all have waivers of TPS requirements due to equivalent experience.