The new law says you must wear a mask unless you have a reasonable excuse not to.
Here is the list of reasonable excuses. Medical evidence would only be useful for a subset of one of them.
Just like the list of 'reasonable excuses' for leaving your home during lockdown, this list is non-exhaustive. There may be other reasonable excuses that are not on the list.
Also, this does not apply to business owners, employees, contract workers, the police, emergency responders, H&S inspectors or council officers. Children under 11 are also exempt.
@msbevvy - I have asthma and am therefore exempt. I fully intend wearing a mask anyway
If you can wear a mask then you are not exempt. It doesn't depend on having a specific condition, just whether you can wear one or not.
The shops will not enforce this because they do not have the authority to enforce it. Only the police, PCSOs and TfL staff have enforcement powers. Only the police can physically remove you.
Full text of the new law is here:
www.legislation.gov.uk/uksi/2020/791/contents/made
The shops are also bound by the equality act which, contrary to popular opinion, has not been cancelled because of the virus. If they refuse entry to someone who cannot wear a mask because of a disability, or demand that people disclose personal details of their disability, they could be taken to court for discrimination.
The government is very clear that it is not necessary to show evidence and the exemption cards they provide are not certificates. They are for people to use to help them explain that they have an exemption. That's all. They suggest a home made sign as an alternative.
www.gov.uk/government/publications/face-coverings-when-to-wear-one-and-how-to-make-your-own