How long did it take a typical capital warship to ramp up to flank speed?

A major limiting factor on rate of acceleration of an oil fired steam ship is how quickly the boilers can be brought to full pressure.

Large ships, such as battleships and aircraft carriers, have multiple boilers, and at cruising speed will only be operating some of them, as running all boilers at reduced heat is very inefficient. To achieve full power, all boilers have to be operating at their maximum pressure, and that means bringing cold boilers on line. This takes some time, as the boiler must be heated evenly and slowly, lest parts of it expand too quickly, which could result in a rupture, and a horrible death for anyone in the boiler room if that happens.

Here is a simple description for bringing a water tube boiler online. Fire for five minutes, wait 15 minutes, repeat until steam comes out of the bleed line instead of air (this usually takes eight or nine cycles), then slowly step it up to full pressure by firing for 30 minutes, and then waiting ten minutes. The whole process can take three to four hours, though it can be rushed a bit if the alternative is sinking, albeit while placing the machinery room crew at some risk of being boiled alive if a steam line ruptures.

If we assume the warship is operating under wartime conditions, and the crew has kept the inactive boilers at least partly heated in anticipation of a change of speed, all boilers could be brought online in around two to three hours.

So, to begin with, the battleship captain must allow a minimum of two to three hours just to get enough steam pressure on all boilers to even contemplate high speed. And then there is the matter of accelerating a mass of 30,000+ tons in water, after all boilers have achieved operating pressure.

This isn't as much of an issue with nuclear powered warships, as the heat is applied to the water in the boiler internally with a heating loop, not externally by first heating the boiler. The water itself acts as a thermal buffer for the boiler, so they can be heated more quickly without as much concern for uneven heating and thermal shock.

The US navy has abandoned oil fired steam plants entirely, replaced with either nuclear plants or gas turbines. The gas turbines are much lower maintenance, and can be brought to full power in a matter of seconds.

The answer to that is difficult. It has to do mainly with technological limitations and fluid dynamics.

In fact, you should ask about time to flank speed, as this would be your only measure of maximum output shaft horsepower (shafts, actually). The power delivered to the shaft is just enough to propel ship at desired speed. For example, HMS Canada could cruise at 14 knots using less than 18% of nominal power, yet reaching 22 knots required her full (nominal) power of over 38k shp (303 RPM). And during trials she exceeded 24 knots, at 52.6k shp (and propeller shaft RPM exceeded 335). See "British Battleships of World War One: New Revised Edition" for more details.

Anyway, the problem at some point is water... Because at those speeds water is... well, hard would be one way of putting it. Another is: when one goes that fast it's not a lubricant anymore...

Another problem is cavitation, which was recognized well after 1960s, and which reduces the efficiency of powertrain.

And last but not least... With the exception of nuclear-powered ships designed from keel as such, ships' designs are optimized for cruising speed (which is not the max speed), and those designs are not efficient at high speeds. There are ways to alleviate that, but obviously older designs would be deficient in that regard too, sometimes greatly. This has to do with fuel consumption, which is not a concern for nuclear-powered vessels. So while it's not in your question, I would recommend to do any comparison on era-by-era basis.

So. By now you should see the problem with your question...

Typically battleships of that size (25-30k tons displacement) required about 50-60 min to reach maximum speed. They could reach maximum power only during max-speed runs, as power produced by the engines must be dumped somewhere, and you literally can't run ship's propeller shaft from start to 330RPM in no-time. It's same thing everywhere - when you want to burn tyres when accelerating in your car there are parts of the transmission system that would prevent engine from sending all it's power to the powertrain... There is a reason why new (that is new when they first unveiled the new generation) Nissan GTR had a caveat in warranty stating pressing "race mode" button voids it as it is likely to damage shaft.

And the numbers from opening paragraphs are - let's be honest - nothing. USS Iowa, when on her trials, reached contracted 32.5 knots, what requried about 212k shp. And she was running light for that. How long it took is nowhere to be found (those numbers are classified), but since she was larger than Canada or Hood (and heavier) it would be definitely longer. For some basic info on British ships see here, for stats on Iowa here. CVNs have on hand even more: Enterprise was rated for 31 knots at 280k shp from 8 nuclear reactors, now Truman has 2 reactors outputting 260k shp.

[It is interesting to see that, for example, Scharnhorst required doubling of shp output to add just 4kt to already high speed of 26kt. This is the "water is not a lubricant anymore" issue at full display here.]

And one must not forget that USS Iowa was built with designed overload of 20%, so we don't really know how fast really she could go. Same thing would apply to most WWII-era ships.

Last point to make: USS Iowa was rated for power output of 158MW. That's slightly more than 80% of the Nimitz Class CVs 194MW. Considering that both are rated at 30+ kts max speed, it shows the problems, limitations and solutions used: that a battleship with a full load displacement of a bit more than half of the 'Nimitz Class CV`'s (58kt vs 102kt) has that much power and is just a smidge faster than a CV?