//I would think medical evidence would be needed only if you claimed a medical exemption and your case reached court.//
No it wouldn't for two reasons:
1. If the legislation for wearing face coverings in shops follows that for on public transport, you will be exempt if you have a "reasonable excuse". A non-exhaustive list of seven reasonable excuses is provided and five of them are not medically related and could not be supported by medical evidence.
2. More importantly, because of the way the public transport legislation is worded (and again I assume the shop legislation will be similar) it would be up to the prosecution to prove that the defendant did not have a reasonable excuse, not for him to prove that he did. Of course proving a negative is notoriously difficult and the legislation is badly framed. It says: "No person may, without reasonable excuse, use a public transport service without wearing a face covering." So, the prosecution would have to prove that the defendant was (1) using public transport and (2) did not have a reasonable excuse. Only if they did that would the defendant have a case to answer. I cannot see a successful prosecution under that legislation as it stands if there is no co-operation from the defendant. If you want I'll expand a bit on this tomorrow and in particular the difference between this legislation and S172 of the Road Traffic Act, which requires the Registered Keeper (RK) of a vehicle to provide the driver's details at the time of an alleged offence unless he is unable to. That has been properly framed to require the RK to explain why he cannot name the driver if he wants to avoid conviction for failing to do so. But for now I'm off to bed.