Firstly, this is a not a free speech issue, imo. This is a public/patient safety issue. For example, would we consider it simply a personal issue, and therefore not a work disciplinary measure, if a pair of teachers published a commentary that was extremely hostile and insulting to, and showing a complete lack of understanding of the children in their care? Should we not be concerned if a pair of oncologists wrote in BMJ RR that their cancer patients were whining fakers who didn't need or deserve chemotherapy, radiotherapy, or surgery?
The NHS is also a public body where it is vital that where known problems are not being properly addressed, that staff must be free to speak out, and therefore the capacity of management to inhibit staff from public communication must necessarily be limited.
The issue in the case of Collings and Newton (or potentially just one of them) is (it appears) that they failed to appreciate that their hypotheses would cause upset to patients, and that they (or at least Newton) used their institution's name without seeking any permissions, as validation of their views. Had they gone to the management and sought agreement on what was to be published, it is very likely the absurdity of their hypotheses and the inevitable offence caused would have been pointed out to them and permissions witheld.
From the NHS perspective, in general the view would likely be that this is a case of severe nonclinical misjudgement and a failure to follow approriate protocols on publishing and using the employing institution's name. The NHS would be unlikely to want to see a loss of highly trained staff in such circumstances and sanctions far short of sacking would be expected. In some circumstances, it's always possible that a disciplinary action is seen a useful way of moving someone on, but staffing reperesentatives would be very resistant to someone being sacked for expression of a professional opinion even if it was ineptly delivered.