It's a tricky one... Your medical record is your data, so as a data subject (legal definition in the data protection act) you can request visibility of the information held about you, and if it is incorrect/missing/etc you can request that it is changed/corrected/etc. But the data controller which is this case the NHS and by proxy your GP surgery also has a legal duty to ensure that data kept about you is accurate. So they have to be able to satisfy themselves that any assertion from yourself about completion of a medical procedure is true.
The issue really here is that the records made available by the system in the US for successful completion of vaccination is not fit for purpose, and any other authority is going to struggle to accept a CDC card as a record of vaccination, not just the NHS. I suspect that this is going to end up causing a whole bunch of pain for those who have been vaccinated in the US.