Whilst I agree, Rounduit, that these are all good/better controls, it is not really relevant to the OP question. The question was whether the driver could appeal under "human rights", and the answer is yes, but he will lose. No-one would take up the case, as it is open and shut.
As was noted in the previous discussion, the driver was not fired due to him not wearing PPE, it was his attitude that cost him his job. He was rude to a customer and refused to apologise. The face covering was the catalyst. Some sites have stupid health and safety rules, but if you don't follow them the customer can exclude you from the site. As long as this exclusion is not based on any protected characteristic (age, disability, gender reassignment, marriage and civil partnership, pregnancy and maternity, race, religion or belief, sex, and sexual orientation), and meets the 3 criteria laid out below you have not had any of your rights unfairly restricted.
At the very loosest of terms the only right I can see that may have been violated would be "the freedom of expression", this is a qualified right. A judge would then look at the three clauses, which have to be met to overide that right.
Is there a legal basis? - Yes, under the HASAWa etc 1974 every employer has a duty to ensure that, so far as is reasonably practicable, the health, safety and welfare of employees are protected. Face coverings are a reasonably practicable way of protecting your employees (and others on your site)
Is there a legitimate aim or justification for the restriction? - Yes, the protection of public health, and/or safety
Is the action proportionate? - (ok this is where we may disagree) A face mask is cheap, effective and easily available (they even gave the driver one). That seems to meet all the criteria to what I would accept as proportionate. Yes, there were other ways they could have done this. But this is the control they chose.
I acknowledge that you are more experienced than I in the H&S field, so I welcome critique from my peers. If I am wrong I would prefer to know and understand that. If you can explain, for example, why you feel face coverings are not proportionate, that may help me understand your point better.