This is 'guesswork' rather than 'knowledge' but I'll have a go anyway:
Lizzie's coronation was in 1953, when Britain was finally starting to see some 'light at the end of the tunnel' in terms of its post-war recovery. (Rationing didn't completely come to an end until a year after the coronation). The government was committed to maximising industrial output, to recover from a devastated economic position. It would have been seen as inappropriate (or even as reckless) to introduce an extra public holiday, in Britain, at that time.
While many Commonwealth countries gave considerable support to the war effort, they didn't suffer the physical damage (from German bombing raids) which affected large parts of the UK. Consequently, their economies were not as badly affected as that of the UK and they were in a better position to declare the Queen's official birthday to be an additional public holiday, without causing any significant damage to their own economic positions.
Chris