What government investigations have been undertaken to determine that there will be no dangerous, cumulative build-up of cancer-causing ethylene oxide in volunteers’ bodies / bloodstreams as time goes on?

On the evidence of the scrutiny and analysis already done – which covered efficacy, robustness, functionality, but didn’t pause to include safety, we’d say the answer is… no:

https://www.gov.uk/government/publications/assessment-and-procurement-of-coronavirus-covid-19-tests/protocol-for-evaluation-of-rapid-diagnostic-assays-for-specific-sars-cov-2-antigens-lateral-flow-devices

The history to this issue:
https://wirralinittogether.blog/2021/03/20/has-your-child-brought-this-nhs-covid19-testing-kit-home-from-school/

Ethylene Oxide Inside Children’s Covid19 Rapid Antigen Lateral Flow Test Kits

Cancer-causing ethylene oxide inside children’s Covid19 test kits. There’s no acknowledgement of any danger, but we’ve had a response from the school to our concerns

Cancer-causing ethylene oxide in children’s Covid19 Rapid Flow test kits. A quick response is in from Public Health England (North West)

https://wirralinittogether.blog/2021/03/30/ethylene-oxide-inside-our-childrens-covid19-testing-kits-a-reply-has-arrived-from-wirral-deputy-director-simone-whites-personal-assistant/