And why is the proportion of deaths to recorded cases so much lower in certain other countries? Presumably it is 1.5% in New Zealand because pretty much all cases got recorded and old people did not get the brunt of it. Why isn't the ration in the UK now somewhere near that?
There is one major difference: New Zealand (and Australia until recently) had very little community transmission, which means there is a very different demographic distribution as to who was infected.
We cannot and should not assume that testing is somehow a magnitude of order lower than it should be without actually doing epidemiological modelling based on who is infected and try to estimate the various risk factors.
The mortality patterns we are seeing in the UK and elsewhere reflect the demographics of who is infected. A high rate of community transmission among the high-risk population can skew the numbers in the way that we are seeing.
Indeed for this (up to July) dataset, roughly 36% of the ~250,000 COVID cases in the UK were been in the 65+ age group, whereas this is only 18% of the overall population. This skew increases as age increases, with 21% of positive cases over the age of 80, who unfortunately have a very high risk of mortality. (With my short search, I could not find more granular data on UK population demographics to do a deeper analysis)
The Geneva Switzerland study showed that the 65+ demographic had ~40 times greater likelihood of mortality than the 50-64 age group.
While it is certain that testing rates have been lower than they should be, given the demographics of who is infected and subsequent mortality rates, it is quite unlikely that true infection rates are ten times the official number.
Perhaps the true "undertest" factor is in the ballpark of 1.5-3 times and this number, like the case mortality rate, will also decrease over time.
Note that while Italy and Spain have similar case mortality rates to the UK, other countries with different demographics of cases have much lower rates.