I think it would be a lot simpler if you picked one country and did everything there. Once certificated, the process of transferring to your home country should be relatively easy.
Lots of Europeans come to the USA for flight training due to the cost, less restrictive regulation, and weather. In ~1988, I took a Dane from zero to private pilot in 31 days in Arizona. He said that the entire cost of his trip, lodging, and training was well under what the cost would have been in Denmark.