After contacting the DHSC several times trying to get proper legislation to the exemption, It now appears they have decided to change the rules
Exemptions for domestic use
If you have received a trial vaccine as part of a formally approved COVID-19 vaccine trial in the UK, you will be able to prove your status through the NHS COVID Pass or with a letter confirming your participation in the trial from your trial site.
See Residents in England who have taken part in COVID-19 vaccine clinical trials below.
If you’re unable to get vaccinated for medical reasons
There are a small number of exemptions for individuals who have a medical reason which means they should not be vaccinated, and in some cases should not test either. Medical exemption status cannot be based on testing alone.
You can apply for proof of evidence that you have a medical reason why you should not be vaccinated and/or tested. If you are eligible for proof of medical exemption, or a carer obtains this on your behalf, you will be able to use the domestic NHS COVID Pass wherever you need to prove your COVID-19 status.
Currently, businesses and event organisers in England that choose to use the NHS COVID Pass as a condition of entry can decide whether or not to allow in people who self-declare that they are unable to be vaccinated or tested.
Businesses can continue to accept people who self-declare that they are medically exempt until mid December 2021, from which point, have to use the NHS COVID Pass to gain entry in the same way that people who are fully vaccinated do. Your NHS COVID Pass will not show that you have a medical exemption.
You can find out more information including how to apply for the NHS COVID Pass to prove that you’re unable to be vaccinated and/or get tested.
https://www.gov.uk/guidance/nhs-covid-pass#vaccine-clinical-trials