TLDR: I'd say (based on the still fairly incomplete research on this) that the playbook was largely the same as a "listicle" and that e.g. by some indexes the UK and Australia had implemented the same kinds of measures up to this past summer, but the timing of various measures was more problematic in Europe, e.g. the relaxations of rules when cases dropped were apparently not as sizeable in Australia compared to Europe, which allowed cases to rebound in Europe in a more dramatic fashion.
This seems to be in line with expert and journalistic assesments e.g.
“Extreme lockdowns work, no question about it,” said Timothy Sly, an epidemiologist with Ryerson University’s school of public health. “This is what China did, what Melbourne did, and it worked. Partial lockdowns just prolong the agony.”
Several practical measures contributed to Australia's success, experts say. The country chose to quickly and tightly seal its borders, a step some others, notably in Europe, did not take. Health officials rapidly built up the manpower to track down and isolate outbreaks. And unlike the U.S. approach, all of Australia's states either shut their domestic borders or severely limited movement for interstate and, in some cases, intrastate travelers. [...]
Almost all public life in Melbourne ended. After 111 days of lockdown, the number of average daily cases fell below five. On Oct. 28, state officials allowed residents to leave their homes for any reason.
Contrast that with e.g. Macron reufusing to put France into a 3rd lockdown with "just" 25,000 cases per day. Or even ordering a weekend lockdown for Paris.
This is difficult to answer because measures have varied over time in both areas (Europe, Australia). For example, right now, Europe has more measures in place compared to Australia, according to one "stringency index" (of lock-down style measures--OxCGRT). However, one salient point with such lock-down measures is that they are most effective when taken early in an epidemic, which is politically difficult. According to the same index, Australia had more measures in place over the summer (and early autumn--well, from a northern-hemisphere perspective) of 2020, when Europe relaxed those measures, probably too much.
That kind of info is reflected in some expert opinions too, e.g. from Nov 2020:
The lockdown seemed an almost shockingly blunt tool when China first applied it in Hubei province on 23 January. But it also proved remarkably effective, and countries around the world took the same approach in the spring, although with varying degrees of intensity.
Europe has had a more science-driven pandemic response than the United States, but unlike many Asian countries, it was unable to avert a resurgence. Instead of using the summer to drive cases down to practically zero, Europe celebrated the holiday season. People seemed to lose their fear of the virus, says Michael Meyer-Hermann, a modeler at the Helmholtz Centre for Infection Research who was involved in drawing up Germany’s lockdown plans. They increasingly flouted rules on physical distancing, mask wearing, and avoiding large gatherings.
It's difficult to quantitatively evaluate this angle, but there was probably more political push-back in Europe against such measures, e.g.:
Not everyone is convinced lockdowns are the answer. On 28 October, the day Chancellor Angela Merkel announced the new measures, Germany’s National Association of Statutory Health Insurance Physicians presented a strategy paper arguing against a lockdown. “We cannot put the entire country, or even a continent, into an induced coma for weeks or months,” said Andreas Gassen, head of the association.
As delayed lockdowns (and similar non-pharmaceutical interventions) are much less effective, even if the measures adopted are ultimately the same in one country (or even region) vs. another, their timeliness matters a lot with respect to their results. There probably are some cross-country studies like that too, but one that is fairly well known was simply across various regions of Italy:
Regression analysis showed a significant positive correlation between the number of confirmed cases before lockdown and mortality up to sixty days (p < 0.001, R2 = 0.57).
(I've omitted the figure on that one, as the paper is open-access.)
There aren't that many studies out on the 2nd wave in Europe yet, but regarding the first one:
Most European countries were able to successfully end the first wave of the pandemic – defined as a two-week incidence rate smaller than 10 cases per 100,000 people. We find that countries in which the virus made significant landfall later in time enjoyed a latecomer advantage that some of these countries squandered, however, by not responding quickly enough and that an early lockdown was more effective than a hard lockdown.
There's also an IMF working paper that discusses in more detail the
nonlinear effect of lockdowns, relative to their adoption date based on a wider panel of cross-country data (see e.g. figure 2.7 in there).
So simply copying the playbook in terms of a list of measures is not enough if their timing is bad. In some sense, Europe "failed to copy" its own earlier measures (at least in terms of timing) when it came to the 2nd wave.
Related, but somewhat of an aside, there have been some attempts to evaluate the relative effectiveness of such measures (and by this I mean against each other, rather than across countries). I don't know to what extent the various stringency indexes reflect/weight those measures relative to their effectiveness.
N.B. There is one paper which clustered the Covid-19 spread curves by the similarity of
their shapes--see figures 3 [dendorgram] and 4 [cluster-colored map]. This produced several clusters, in which Australia is similar to China's in term of R(t) shape, but Western Europe is another cluster [with Canada]. This however is just a visual way of stating the obvious that Europe experienced a worse 2nd wave. This paper actually distinguished Central-Eastern Europe as its own cluster, as the the latter region saw a smaller 1st wave, but a higher 2nd one relative to Western Europe. (That last point was also raised in another article.) Alas this paper mostly argues from the observed curve of the spread (rather than analyzing the measures themselves) that
The first wave started when there were no interventions and extensive winter tourism in Europe. We never returned to those favourable conditions for virus spread during the summer (i.e., R never returned to the high values of late February), so if Europe had imposed the same restriction in the fall as in the spring the second wave could not have been more severe than the first. The fact that it did grow stronger in terms of incidence rate, reflects the fact that interventions in the fall up to mid-November have been very weak, i.e., we have seen the effect of intervention fatigue.
Somewhat related to that, such "fatigue" was not unique to Europe. Japan for example was discussed in that regard too, having experienced 3 peaks of Covid-19 cases, each one greater than the previous.
There is thankfully one paper which did try to account for the stringency of the measures relative to the level (actually point in time on the growth curve) of the outbreak. Unfortunately, it doesn't feature Australia among the countries compared in its abstract (although it does show up in a table), but nonetheless, even the abstract is informative:
Accounting for response timing, China
imposed the most stringent restrictions, while Sweden and Japan were the least stringent [...] Timing
was fundamental: governments who responded to the pandemic faster saw greater reductions in
viral transmission (p = 0.013), but worse decreases in GDP (p = 0.044). Thus, response stringency has
a greater effect on GDP than infection rates, which are instead affected by the timing of COVID-19
Based on that data, which includes various measures of comparison, e.g. both the peak and area under curve for OxCGRT, up to July 2020, there wasn't much of a difference between Australia and e.g. the UK in terms of measures, even though there was surely a lot more difference in terms of cases... So perhaps this "stringency index" isn't too explanatory. (Actually, the paper itself discussed the issue that the OxCGRT may be "too coarse".)
This is also lackluster as the statistics computed in that table are cumulative and only include the 1st wave in Europe. (Apparently there is some kind of "publishing fatigue" too as it's hard to find papers that cover the 2nd wave in Europe. Perhaps they'll appear in the 2nd half of this year.)
However, there is one thing which is noticeable (e.g. in re Australia vs. UK) in the (next set of) comparative graphs in which both stringency and case curves are plotted, fact which isn't too apparent from the cumulative statistics of stringency alone: Australia persisted with the measures even in the "lull" when there were few cases (as did China, for that matter):
Alas this data is somewhat truncated (like I said) by the time frame considered. It would surely be interesting to see this kind of analysis with at least 6 more months added, i.e. end of 2020, especially to cover/explain what happened in Europe's second wave, relative to other countries.
Perhaps Czechia is illustrative (albeit the worst case) of the problem of bad timing of decisions in Europe. The story is a bit too long to fully recount here, but in a nutshell: after mostly avoiding a first wave in the spring of 2020 due to fairly strict measures, a combination of political factors (i.e. elections) and social factors (backlash against the 1st wave measures that were seemingly unnecessary) led to 3 waves/peaks: one in Nov 2020, one in Jan 2021 and a 3rd in March 2021. The expert opinion regarding this latter sequence seems to be that Czechia waited too long before instituting measures and also relaxed them too quickly, as well as failing to account for the increased transmissibility of the new strains of Covid-19.