There is an interview here on this site between Ajahns Sujato, Brahmali, and Jim Tucker where he (Tucker) says this as well:
Ian Stevenson used to say the word proof should only be used in mathematics. In science it’s all about evidence, not proof. So what is the evidence? And what is the best explanation that the evidence provides? It’s not the same as proof. And sometimes what appears to be the correct explanation is ultimately shown not to be.
But, regardless, this is certainly evidence that some children have memories from a life that occurred in the past. Now the most obvious explanation on the face of it would be that it’s reincarnation, that the child’s memories came from a life they’ve experienced before. …
I think ultimately, in order to convince the open-minded and intelligent, we would need to incorporate these cases into an overall understanding of reality, because they obviously fly in the face of scientific materialism. The general public believes all kinds of things actually, but as far as the reigning intellectual paradigm is concerned, it’s scientific materialism. If it’s gonna happen, I think it’s probably gonna come through physics. If consciousness is to be considered a separate entity from physical matter, a new theory would need to incorporate most of what we already know to be true. But I’ll probably have to wait until my next lifetime, or the one after that, to see that. But I do think it’s possible that ultimately it could come about.
I would like to take a look at this if you happen to be able to locate it. I did a search of BBC for Ian Stevenson as well as Jim Tucker but turned up nothing.