The price was not revealed to the BBC, but is available on application.
When the site first opened it was suggested up to 1,500 jobs would be created, processing up to 300,000 tests a day, but this never happened.
Temporary staff employed at the site were let go earlier this year, with the last day of testing on 16 January.
Mr Western said he was “furious” the building was being sold and that the government had “misled” the public about its long-term future.
“The real concern to the public is that this was a huge investment by the government,” he said.
“We were told that even beyond the pandemic it would have use, a function as a diagnostic testing centre for cancer, cardiovascular and other metabolic diseases and here we are such a short period after and they’re already trying to dispose of it on the quiet.”