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The treaty in question is the Copenhagen Convention of 1857, which is still in force. The text, which took some finding, can be found starting at page 1301 of
Hertslet.
According to article 1, "no Vessel whatever shall henceforward be subjected, under any pretext, to any detention or impediment whatever, in the Passage of the Sound or of the Belts." That would seem to include warships as well as merchant ships, and Wikipedia says this is the case.
The treaty is post-POD, but I'd be very surprised if there weren't something similar in TTL. If you look at the list of signatories, it's basically everyone with any shipping interest in the Baltic, and it's pretty clear that they all ganged up on Denmark to open the straits in return for a cash payment. The same thing would happen in TTL as northern Europe becomes more industrialized and the volume of shipping increases - if Denmark insists on keeping the sound dues, it'll end up on the losing end of a trade war.
So I'll assume that the treaty exists in TTL, plus or minus a few details, which means that it would be legal for either side's warships to pass through the straits.
Um, wow! I wouldn't have guessed that internationalization went quite that far!
Because of course it opens up, in theory, the horrible spectacle of two navies of comparable power meeting, and duking it out, right there in the Straits themselves--which is to say, right in the middle of Denmark itself

--and the Danes legally speaking being restricted to spectators, wringing their hands and perhaps suing for damages for stray shots that happen to devastate their towns, cities and farms!
Not a very realistic prospect here, but say Russia and France had in some third timeline gotten strongly at odds with each other somehow and both built up large navies of comparable power.
Or, I'm not sure why this didn't happen OTL during WWI. The British not wanting to provoke the Danes into joining the Central Powers and perhaps persuading Sweden and Norway to join as well, treaty be damned?
As you say, Russia probably won't exercise this right because it would be suicide; they'd probably concentrate on achieving local superiority within the Baltic and keeping the Gulf of Bothnia open for trade with Sweden. The question is whether the British and North Germans can spare enough warships to close the gulf, or if they're spread too thin to do so in the face of the entire Russian Baltic fleet (especially since the German warships will be needed for home defense).
The Danes will let the ships pass and say as little about them as possible, lest they compromise their neutrality - Scandinavia is indeed neutral.
Well, all my speculations about the possibility of the Russians making the Baltic and Gulf of Bothina a Russian lake depended on my notion that the Danish straits were closed to warships and Kiel canal doesn't exist yet. (I assumed the Germans made Kiel canal OTL because they anticipated trouble going through Denmark, I didn't realize that the trouble would strictly be the RN preempting them by loitering there).
If the RN can move freely into the Baltic, I'd think they would bring in whatever force necessary to shut all Russian shipping, naval and merchant, into their ports near St. Petersburg. Russian trade with Sweden would practically be restricted to their logistically practically useless land border, because not only would the British want to free the North Germans of the distractions of being raided, they'd want to bottle up Russian trade and what naval power they had as tightly as possible.
In fact if the Danish straits are open, that helps answer my question, "why hasn't the RN pressed the French navy very hard off France itself yet?" If in addition to trying to police the world's shipping lanes they need to interdict French traffic across the Med and hold the Russian navy at bay in the Baltic--oh, and reinforce the Dardanelles too and try to hold as much of the Black Sea as they can, again penning the Russians into their ports there--well I guess then it would seem, the mass attack to break the core of French naval force defending the French Atlantic coast could wait.
After all such a battle only benefits BOG through attrition, unless the British are prepared to follow through with landings on the French Atlantic coast--and with what men? What the British can offer by way of Tommies and colonial troops they are currently sending to Germany; they can't dream of opening up another front by invasion until much later.
To be sure, I still think that neither British nor French admirals would be content to just glare at each other across the Channel; both will be itching to see how well they can do against the other. I presume the French have had all kinds of ingenious theories and nifty devices about how they can turn the tables on the mighty RN, at least in the matter of defending their own coasts and opening up wide breaks in the British blockade; the British would like to find out the worth of these newfangled notions almost as much as the French would. Hence my notion there would be a big clash quite early on--and perhaps it turns out the French ideas do have some merit after all, enough to keep the RN from just sinking every French hull in an orgy and then seeming to have no choice but to land what ragtag land forces they had on the defenseless coast.
I'm no more a naval scholar than you are and have no interest in a blow-by-blow bit of battle porn, but I do wonder what measures desperation combined with ingenuity might have been adopted by the French.
Torpedo boat squadrons, lacking range to be a threat on the high seas but capable of turning the tables, or holding their own, close to French ports, perhaps?
There would almost certainly be shore batteries bristling all along the French coasts. I don't know what degree of such guns the French economy could have supported, but whatever they could afford, they'd be there.
With spotters based in balloons, for long lines of sight. Especially if the French went in for some superguns, the way the Germans did OTL.
I pooh-poohed the idea anyone would be using submarines of any kind, but one possible place they might exist is in close defense of harbors; electric battery powered small subs might serve a useful function there, basically supplementing the shore guns and torpedo boats with stealth.
All of these ideas are suggestions of what the French might have done to keep a British landing off their shores and hold British gunships out of shelling range. I hope someone can critique them, and suggest what the French might have hoped to use to break out of the blockade and regain freedom of the seas.
In particular, they have to be worried about how they are going to keep communications open with North Africa; if they can keep ships going between Marseilles and Algeria, the rest can be overland, but can't the British put a stop to that in short order, unless the French have some clever trick up their sleeves?
You mean, aside from the fact that I know jack about late 19th-century naval tactics, and that I doubt I could write a convincing naval battle without considerably more research than I've thus far done?
I'm not looking for battle porn; I'm interested in the technologies and the outcomes.
...I'll assume that, thus far, the French fleet has suffered individual defeats but the RN hasn't been able to gather enough strength in one place to deliver a knockout blow, and that the North Germans - who aren't really threatened by sea, and who need the RN to help protect their merchant shipping - have higher priorities. I'm more than willing to be convinced otherwise, though.
Well, I guess I've come up with my own excuses why the RN might have held off. I'm not convinced entirely; I'd think there would have been a big battle early on, and if then French tricks turned out to be good, at least for purposes of making any close approach to French shores very costly, then they'd back off, because until the British accumulate an invasion port of decent size there' no point in coming in that close. Conversely if the French had some magic trick that let them break out of the blockade, they'd use it right away and you couldn't avoid talking about it, so they don't have that. So it's a stalemate for now, but I think they'd prove it to each other first before backing off and waiting for other factors to line up.
Given the constant drain of troops piecemeal into Germany, it may be a good long while before the Admiralty has to face the question of how to break through the French defenses, and whether that's a good thing to try or not.
I wonder if, if that is in the cards eventually, the invasion force the RN opens the way for will include lots of Ottoman troops?
It could also be, no extra front will ever be opened by a landing; that the strategy is to stay with the fronts they've got, to push the French out of Germany and to beef up the Italians so as to invade France from the south.