The NHS system was introduced in the UK in 1948. I think Lord Devlin is thinking mainly about the system before the NHS was introduced (which continues after 1948 to a small extent). Before 1948 doctors would charge patients a set fee for a consultation but if the patient was poor the doctor might decide not to accept a fee. Hence no consideration and so no contract.
Although Lord Devlin is writing in 1963, and the events giving rise to the case occurred in 1958, the law he is considering was developed mainly pre-1948 and Lord Devlin himself, his fellow judges, and the lawyers would be familiar with the pre-1948 practice of doctors. Indeed when the NHS was introduced about 5% of general practitioners (non-hospital doctors) stayed outside it. NHS doctors were not employed by the NHS - they simply had a contract to treat registered NHS patients and were, of course, free to carry on private practice as well. The 5% were doctors who only did private practice. Their patients would have been those with higher incomes which, no doubt, included judges and barristers.
The situation prior to 1948 is described in this article from a UK government website:
Payment for medical treatment depended on occupational status, class,
gender, and age. Many working men possessed health coverage under the
1911 National Insurance Act, which provided access to a ‘panel’ doctor
for a contribution from their weekly wages. ‘Dependents’ (wives and
children), however, were not covered by the 1911 Act and had to pay
out of their own pockets to see a doctor (typically 3 shillings and
sixpence in the 1930s). As a result, women and children faced some of
the harshest barriers to medical care in the interwar period. As
national insurance depended on job status, the long term unemployed
also often struggled to access affordable healthcare. Surprisingly, so
could many members of the middle class. Many found paying private fees
difficult and were disqualified for coverage under the 1911 Act if
they earned above the wage limit.
With the introduction of the NHS, nearly all those on low incomes would no doubt have registered with an NHS doctor and if their usual doctor was not in the NHS they would have switched to a doctor who was. But there would have been many people who could afford to pay privately who would continue to do so particularly if their usual doctor, who they wish to continue to use, was in the 5% who were not in the NHS.
It is not difficult to envisage situations where a patient who normally has no difficulty paying privately might have difficulty paying. You can imagine a situation where someone off work with a serious illness is visited by the doctor on successive days during a week. They may be able to afford to pay for the first two visits but run out of money and the doctor might then choose not to charge a fee when making subsequent visits. So even after 1948 it is possible to imagine non-NHS doctors deciding not to charge a fee on some occasions.
The case, of course (Hedley Byrne v Heller) is not a medical negligence case at all - Lord Devlin is just using the example of medical negligence to show that the distinction, hitherto made between what can be recovered in tort and what can be recovered in contract, lacks common sense.
The gist of Lord Devlin's argument is that in a situation where someone with a serious illness is visited by the doctor on successive days during a week, sometimes paying and sometimes not, if the doctor negligently advises the patient that they will never work again and the patient resigns their employment only to later recover completely so that they suffer economic loss due to not being able to go back to their old job, it would be anomalous, Lord Devin would argue, if the success of their subsequent court claim against the doctor hinged on what day of the week the negligent advice was given on.
It is not difficult to guess why Lord Devlin does not use the example of doctors providing treatment on the NHS (where the doctor has a contract with the NHS to which the patient is not party so that the patient can only sue in tort) compared to doctors providing medical treatment privately for a fee paid by the patient (where the patient can sue in contract). First of all there are many practical differences. A general practitioner doctor who does NHS work will have an overall contract with the NHS which generally requires them to provide a service to a certain number of registered patients. The NHS does not pay a separate fee for each thing the doctor does (though there might be incentive payments offered for, for example, every vaccination given). A doctor seeing a private patient can decide on any number of contractual arrangements - fee per consultation, or a overall annual "plan" for a set fee perhaps with charges for "extras". The whole thrust of Lord Devlin's argument is "why should the law make a difference between A and B when the practical differences are trivial?" Obviously that argument loses much of its force if the differences between A and B are in fact not trivial. You can try to explain the practical differences away saying that they should not matter but such an argument is less attractive than if you use the example of a non-NHS doctor where the difference really could be quite trivial.
Secondly, if Lord Devlin had introduced the NHS system into his argument it might invite a public policy counter-argument. If the law allowed greater damages in tort that would mean more money spent by the government on the NHS and less money for other public services. So, it could be argued, the distinction between tort and contract damages might not be so silly after all. That would not have helped Lord Devlin's argument.
At the end of the day the case Lord Devlin is giving his speech in (Hedley Byrne v Heller) is a case on negligent bankers' references. It is not itself a medical negligence case so when using medical negligence as an example he does not need to refer to medical practice as it typically is at the time he is giving the speech if medical practice pre 1948 (still within living memory and covered in law reports, and to some extent still continuing in the case of non-NHS doctors) serves his argument better.
Note on the word private
Some commentators on Law SE have suggested that the fact that Lord Devlin talks about private patients must mean that the doctor he is describing must be a post 1948 NHS doctor. They reason what before state-run healthcare existed all patients were private patients so that the private/non-private split would not have made sense prior to 1948. I believe this to be based on a misunderstanding of the different meanings of private
The word private means different things in different contexts and, to an extent, in different English-speaking countries.
In England if you were having a political/economic discussion about the role of private enterprise and what matters should instead be covered by government agencies you would refer to an insurance company as a private company. But if you are talking about shares then you would describe an insurance company as a public company if its shares are traded on the stock exchange or a private company if not.
Another use of private is this. If someone pays for a service directly and it is the kind of service which is often covered by insurance (such as crash repairs) the person who pays directly is often referred to as paying privately to distinguish that from cases where an insurance company is paying.
This article from a UK government website demonstrates that the expression paying private fees was in use before 1948:
Payment for medical treatment depended on occupational status, class,
gender, and age. Many working men possessed health coverage under the
1911 National Insurance Act, which provided access to a ‘panel’ doctor
for a contribution from their weekly wages. ‘Dependents’ (wives and
children), however, were not covered by the 1911 Act and had to pay
out of their own pockets to see a doctor (typically 3 shillings and
sixpence in the 1930s). As a result, women and children faced some of
the harshest barriers to medical care in the interwar period. As
national insurance depended on job status, the long term unemployed
also often struggled to access affordable healthcare. Surprisingly, so
could many members of the middle class. Many found paying private fees
difficult and were disqualified for coverage under the 1911 Act if
they earned above the wage limit.