Re: "We all want to see hard proof—" No, we don't.
The Five Eyes stuff is far more complicated than it may at first sight appear. The UK has _at least_ two sites that the US intel community likes a lot, and which (for various technical reasons) are not easily replaced without a lot of impact to existing programs, so figure that as a multi-hundred-billion dollar relocation cost. Technically, one of the sites might be replaced with a slightly sub-optimal alternate in either one of two European countries, but the other would be harder.
And anyway, "Five Eyes" is not an automatic thing. There's "REL AUS/CAN/NZ/GBR" and there's e.g. "REL GBR", etc. So what the US is threatening is that not as many goodies would be shared, not that nothing would be shared.
At the end of the day, this is all purely political: if GCHQ says the risk is technically manageable, then it's probable that the NSA would say the same (technically). Therefore (from a UK perspective) the US can rattle its sabre all it likes, and when the political climate shifts, they'll "reach an accomodation". Again.