A Type 45 destroyer is almost 10000 tons though, a WWII destroyer would have been 2 to 3000. It’s probably difficult to come up with an accurate cost comparison, but I suspect the difference is even greater than for tonnage.
[1] 1920s and early 1930s, for those who don't know their naval history.
[2] HMS Dreadnought being launched in 1906 and essentially making all existing battleships obsolete on launch.
Edit: Of course it would lose, I had forgotten about submarines!
Even pre-missile refit Iowa (yeah, non RN) would absolutely trash the 1918 Royal Navy. Automated radar controlled fire control is ridiculously good compared to the fire control solutions they had in 1918.
The only saving grace for Royal Navy would be that a single Iowa would not have enough ammunition.
Or really, who knows what real modern combat between naval powers would look like. Let's hope we never find out.
Each destroyer and frigate has 8 Harpoon anti-ship missiles, for a total of 152 ships down. The 4.5" gun (http://www.navweaps.com/Weapons/WNBR_45-55_mk8.php) probably doesn't have the heft to penetrate the armor, although you probably wouldn't want to be in the superstructure of a 1918 era ship.
According to https://www.naval-encyclopedia.com/ww1/royal-navy-1914, in 1914 the RN had 82 battleships (pre-dreadnoughts, dreadnoughts, and battlecruisers (dammit, those are cruisers, not battleships)), 136 cruisers of various flavors, 142 destroyers, and 80 torpedo boats. (The WW1 RN has a significant advantage in torpedo numbers, anyway.) During the war, they built 21 battleships, 32 cruisers, and 232 destroyers. (And I am too lazy to add up the losses.)
So on the theory that 1918 battleships and cruisers are immune to 4.5" guns and that if you throw enough shells at something, you'll eventually kill it, in the surface warfare front, I'm going out on a limb and give 1918 the nod. The modern Royal Navy doesn't really look like it's set up to take on a large surface fleet.
On the other hand 2019 has the Queen Elizabeth with 24-36 F-35Bs and 14 helicopters. Each can carry, say, 6 AGM-158 air-to-surface cruise missiles, for another 216 hits assuming that 1000lb warhead can penetrate armor. I'm sure they'll figure something out. (Worst case, they use the helicopters to ferry marines over and take the ships the olde-fashionede way.)
We won't talk about WW1-era aircraft carriers (or planes for that matter).
Submarine-wise, 1918 had 80 + 156 submarines, including the M class "sub-cruisers". (If you mounted that 12" gun on something modern, we could talk armor penetration.) Unfortunately, most of the 2019 RN helicopters and surface ships are built around submarine-killing and 1918 submarines aren't going to make too difficult targets.
2019 has 4 ballistic missile submarines, each with 4 torpedo tubes (but no indication of how many torpedoes, thank you Wikipedia) and 16 missile tubes, and 6 fleet submarines, with 6 tubes and up to 30-38 torpedoes.
Approximately 300 torpedoes would be a problem for the 1918 fleet. (But it would likely take more than one torpedo to sink a battleship.) However, consider that fleet command and control is a big problem in 1918; they're passing orders with signal flags and lights and attacking in battle lines with a thousand yards or so between ships. With them bunched up like that, those ballistic nuclear missiles begin to get appealing, and 2019 has plenty of 100-500 kiloton warheads to throw around.
Overall, yeah, 2019 has an extreme edge.
I wouldn't underestimate the morale hit from having entire fleets blowing up with no enemy in sight, either.
Shoot something like it to a battleship and you make a tiny hole and maybe break a laundry machine. Sure with enough shells you could disable it but that requires a ton and more of shells.
To damage a ship you don’t only need to penetrate the armor you also need to deliver substantial amount of explosives inside said armor.
Don’t get me wrong. The fleet would be wiped out. But against a BB puny 4.5” does nothing, except maybe when delivering incendiaries to the superstructure. Modern fire control allows it to absolutely wreck all destroyers and smaller though combined with autoloaders.
EDIT: More importantly in order to even have that reasonable penetration chance with 4.5” gun you need to be within few kilometers. And at that distance the 1918 RN is really dangerous. Ideally stay farther than maybe 20km.
Not sure they would. The standard RN small ship gun in WWII was 4.7". The standing orders if a Captain of a small ship sighted a German battleship was to load armour piercing rounds and aim for rangefinders or radar antennae, there wasn't any expectation of doing much damage to elsewhere on the target. His relatives would get given his posthumous VC some time later.
There are youtube videos of using modern torpedoes on ships. One should be enough to sink anything, particularly if it detonated under a battleship magazine.
You could also compare the effects of early guided bombs on battleships [1], they were comparable in weight and warhead to modern guided bombs and to things like the Harpoon missile.
The real change would be the ~20 rounds per minute rate of fire plus modern fire control.
Unfortunately, we seem to be incapable of actually producing such vessels, as too many of the people advocating for naval expansion are advocating for more of the big, over-capable ships (often for selfish reasons).
Sigh.