It is not mentioned in the movies or any of the shows I have seen. Cochrane is mentioned only in passing a few times and actually has a short cameo in Enterprise. At the sake of being chastised, the only explanation that makes sense is that Cochrane may have stolen or otherwise illicitly acquired the necessary materials; he didn't actually pay for anything.
You bring up valid points and all question Cochrane's funding, but the theft theory easily answers them.
It is clear that Cochrane was a bit of a scoundrel. He was a drunk and self absorbed. He even admits in the movie that he only made the warp engine for the money (so that rules out stealing money to buy the materials). He was likely going to sell it to the highest bidder, despite what they might do with it.
Considering the timeline puts Cochrane in a war ravaged world, where most militaries and governments have collapsed and devolved into "faction" rivalry, and also considering his pre-warp character, it seems likely to me that Cochrane acquired the necessary materials illicitly. He either stole them or salvaged them from abandoned government buildings, machines, or vehicles. We are given very little history about Cochrane before the movie timeline. Considering he was smart enough to design and build a warp engine it is not a stretch to say that he previously worked for one of the governments on some high level stuff (like antimatter bombs) where he would have had access to these things. This might also explain why he was in the middle of Nowhere, Montana; he was hiding out.