The following is from Part 1, 1 (4) of the Transport Act 1962 as originally enacted and available on .legislation.gov.uk. The British Railways Board shall consist of a chairman, a vice chairman, or two vice chairmen, and not more than sixteen nor less than ten other members; the chairmen and other members of the Board shall be appointed from among persons who appear to the Minister to have wide experience of, and to have shown capacity in, transport, industrial, commercial or financial matters, administration, applied science or the organisation of workers, and the Minister in appointing them shall have regard to the desirability of having members who are familiar with the special requirements and circumstances of particular regions and areas served by the Board.’
Part 2, 2 covers the Regional Railway Boards and, although it does not explicitly say so, it is to be inferred that the same guiding principles applied to them as to the main Board. The Act does give the official name as London Midland Railway Board.
I am fairly sure that Scottish Board, at least, had several members of the ‘great and good‘. Some of these were inclined to devote a lot of effort to their hobby-horses.