Approval, Plurality, and score voting are the simplest voting methods to count. That causes the time-delay between election day and winner-set-announcement to be short. But the delays can be much longer for harder-to-count election systems like Instant Runoff Voting (IRV).
Fargo, the first city in the USA to enact Approval voting, held its first approval voting election on 9 June 2020 (7 candidates) and announced the two winners on the same day. (And score- and approval-style political polls are commonplace in the USA, with their results usually announced within a few days.)
Now let's compare the time delays between election day and winner-set announcement for the last 4 federal elections in
We see that Canada found its results faster than Australia every time (typically 9 times faster), despite having more MPs, land, and population, less GDP/capita, and older technology. We also see that Germany performed comparably to Canada (despite having more MPs and voters than Canada and Australia combined, as well as more states than either), and also outsped Australia every time. Although the MMP system might naively seem more complicated than plain PV, if you think about the pheneomena which slow winner-determination in the real world, you should realize MMP's real-life speed should be comparable to PV's.
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(*) Australia 2013 was unable to determine Western Australia's 6 Senate results, hence on 18 Feb 2014 (164 days after!) voided WA's senate-election, then held a new one 6 April, with final result announced 7 July (245 days after the original election). Canada announced its 2015 results on 20 Oct, but then did 4 recounts with their results announced 26 Oct. In Canada 2011 there were 4 recounts ordered on 5 May, whose results were announced 14,17,23,27 May; then one was challenged, but the court ruled it valid on 25 Oct.
So evidently simplicity matters a great deal in the real world.
Statistical significance of that: Under the "null hypothesis" that IRV and PV were equally easy to count, the chance that, by pure bad luck, among the 8 House elections considered here, the 4 Australian IRV ones would happen to be the 4 longest-to-count, would be 4!2/8!=242/40320=1/70. I.e. the confidence that IRV takes longer exceeds 98.5%.
And time delays can have severe consequences. Consider the USA presidential elections of 2004, 2008, 2012, 2016, and 2020, respectively won by Bush, Obama, Obama, Trump, and Biden. In all the first four, the winner become clear enough that the 2nd-place finisher conceded defeat on the day after election day.
But in 2020, that changed. There were unusual delays, in large part caused by new voting procedures (massive vote-by-mail) put in place due to the COVID-19 pandemic. This caused it to take 4 days, not 1, before the results became clear enough for the Associated Press, Fox News, NBC, and CBS to call the race for Biden. (CNN called it after only 3 days.) Biden won the popular vote by a margin of over 7 million, and the electoral college by 306 versus 232 (margin 74). Both these margins were greater than usual, although not tremendously greater.
Importantly, on election night everybody knew there were going to be delays.
Trump on election night (3 Nov.) falsely declared victory in Pennsylvania, Georgia, and the entire USA. He later also falsely declared victory in Arizona. Trump also later declared there had been a "major fraud on our nation" (e.g, in Michigan) and said he would take the case to the Supreme Court (they dismissed it without hearing it). Trump lawyers brought over 70 court cases trying to stop election counting, overturn elections etc throughout the country, almost all of them dismissed due to tremendous lack of evidence of anything. In particular, Trump attempted (unsuccessfully) to get the Republican-controlled Michigan State Legislature to overturn Biden's victory in Michigan, after e.g. suing to stop Michigan's count because (he claimed) a GOP-observer had been denied access. Trump's suit was dismissed since the counting already was finished, and also since (the judge noted) Trump's complaint failed to state "why", "when, where, or by whom" the election observer was allegedly blocked. Trump also sued (also unsuccessfully) in the Nevada supreme court to nullify Biden's victory in Nevada and simply declare Trump the winner and his electors elected. He also sued (also unsuccessfully) in Pennsylvania to delay certification and have millions of votes invalidated. Trump also tried phoning his latest Attorney General (Rosen) on 27 Dec. 2020 to urge him to have his Justice Department declare the election "corrupt" then "leave the rest to me" and the Republican congressmen. On 6 Jan. 2021 Trump then incited a riot by his most rabid supporters, which stormed through the capitol building, delaying congress' pro-forma certification of Biden as the winner based on the compiled 50 state results. The riot caused a lot of damage, 5 deaths, and many injuries, but could easily have been far worse.
As of July 2021 Trump still claims he won, indeed that he expects to be "reinstated" as President somehow on 13 August 2021; and he never conceded defeat either USA-wide or in any of those states. The lesson: delays matter.
On 5 Nov. 2019 New York City enacted IRV5 for its "primary" elections (most importantly for Mayor). This made New York the largest city ever to employ IRV. New York then held its first IRV primary election on 22 June 2021. The NYC-BOE announced "preliminary" (since omitting 125000 absentee ballots) counts on 29 June. However it retracted them the next day because it realized the prior announcement had been hugely erroneous! It ended up taking 14 days for the results to get clear enough for the Associated Press to recognize a winner. That's 6 times the average delay in NYC's preceding four mayoral primaries in 2017, 2013, 2009, and 2005 (all PV) whose winners were recognized (e.g. loser conceded) after 1, 6, 1, and 1-day delays.
Statistical significance of that: The chance that the reason New York's IRV primary took longer to count than its last 4 PV primaries combined was just "bad luck," certainly was ≤1/5. If we combine this with our previous significance calculation (for Australia vs. Canada) we find combined confidence>99.7% that IRV takes longer to count than PV.
Greater confidences would arise if we were to use, not merely the fact the IRV elections took longer, but also how much longer, under some assumed model of time-delay distribution. For example, if the time-delays were assumed to be Gamma(2) distributed, then this confidence should exceed 99.99%. And if the Australian Senate (not just its House) delay-data were used, that also would yield increased confidences for appropriate statements.
We also could note that, e.g, for every pair (A,C) of Australian and Canadian elections listed, the Australian took over twice as long, except for 1 out of the 32 possible pairs; this also was true of the 4 NY City (IRV, PV) pairs. In all that is 36 election pairs, and in 35 of them the rank-order ballot election took over twice as long as the PV election, while in the remaining 1 case, it took "longer, but not twice as long."
More data: The following do not seem to yield any statistically significant conclusion, but it perhaps has some interest to compare the last 6 London Mayoral elections (2000, 2004, 2008, 2012, 2016, 2021), all using the "Supplementary Vote" (SupV), versus the last 5 in Los Angeles (1997, 2001, 2005, 2009, 2013, 2017), all done via Plurality plus (if needed) separate top-2 runoff (T2R).
The SupV system, now used for all mayoral elections in England and Wales, was the brainchild of Dale Campbell-Savours, then Labour MP for Workington. Under this system voters express a first preference and (optionally) a second. Everybody is eliminated except the top two finishers based on top-preference votes only. Then any votes for eliminated candidates transfer to that ballot's second choice, whereupon the candidate (among the final two) with the most votes wins. SupV can approximately be regarded as an "instant" version of T2R.
The delays between London's election day and the announcement of the winner, were 1,0,1,1,3,2 days, averaging 1.33. The delays between LA's election day and the announcement of the winner (or of 2 winners, in the 3 cases where nobody got 50%, necessitating a later runoff between them) were 1,1,2,1,1,1 days, averaging 1.17. LA's mayoral runoffs, when needed, were held about 2 months after.