... A healthier heir for the Czars could make the difference between OTL's Red Revolution and a shift to a constitutional monarchy...
I think that given the same domestic situation in Russia in 1917---minimal butterflies--it would not matter how healthy the Tsarvich is; given the balance of power of the revolutionary classes (that is, the vast majority of Russians) the monarchy falls and some sort of radical republic--again with minimal butterflies, the Bolsheviks who would have no truck with any form of monarchy, but even the SRs would probably reject it too.
Now the vast majority of comments seem to assume maximal rather than minimal butterflies. Certainly if Victoria, a woman of just the same personality who happens not to be very fertile for reasons that don't affect the formation of her personality, takes the throne in a context identical to OTL, we still would expect serious butterflies to start taking effect within the next couple decades.
By strong butterfly theory that says that any small change must cascade into everyone born within a decade or so of the change and after being different people, we'd have to game out a completely different Europe--insofar as great events are determined by individual personal whims rather than deep underlying forces. German unification under the Hohenzollerns for instance--was that a product of whimsical choices made by the increasingly divergent Prussian kings, and of course their counterparts across Europe, or was that something that (at least given decisions already made before Victoria's coronation) was set in stone already, due to the distribution of power that in turn more or less radiates from developing industrial power distribution and the imperatives of policy makers responding to those developments?
Taking a broad-brush deep forces point of view, the ambitions of whoever runs Prussia would seem to indicate a similar course; the 1848 revolutions happen despite butterflies coming out of Britain, which sets up the political landscape for German unification, deep level conflict between a post-unification German and French pair of regimes (that is, even if France does not go Republican, the conflict and basic levers of power and their shifting balances remain the same); the British position evolves similarly, we wind up with the Entente/Central powers polarization, the Great War happens within a couple years of OTL, with similar outcome....all this is necessary to even raise the question of a crisis in the Russian monarchy whereby an armed and very discontented mob is facing the tiny clique of pro-monarchy old regimists who alone even care to continue the Romanov dynasty.
Of course one might also have the broad brush pattern evolve with subtle changes such that there is no February Revolution, say because the Tsarist conduct of the Great War is not perceived as quite so bankrupt; perhaps the women of St. Petersburg are not holding demonstrations because they aren't so desperately hungry due to a more effective management of the war economy and more fair rationing, so the revolutionary crisis does not quite come to the breaking point and instead the Romanov monarch (someone quite different from OTL Nicholas II) finds it expedient to introduce a moderate constitutionalism. But why then? More likely a moderate constitutionalism was introduced earlier, and helps explain somewhat better Russian performance in the war. Then again who exactly is the constituency for moderate, conservative-liberal constitutionalism in Russia anyway? Russia has a different class structure than the western European states. Perhaps that too has been butterflied in the sense that earlier Tsars introduced something like the Stolypin reforms earlier, more gradually, and this somehow led to a stronger agrarian middle class--prosperous peasants--who more or less coordinate interests with a stronger urban/industrial middle class (technicians, foremen etc from modest background who win a higher degree of social respect from the propertied classes proper). But this means that the old propertied classes will have lost substantial grounds, at least in the countryside--can any Tsar, no matter how powerful and enlightened, undermine those classes without provoking dangerous levels of unrest among the well-off? In Russia I would think even if the upper stratum of peasants and workers are substantially better off, this might not defuse the mass unrest of the majority who would be no better off than OTL and quite possibly perceive themselves as worse off. The revolutionary crisis might come much earlier. The proletarian and low peasants are not necessarily foredoomed to win, but the facade of democratic progress would be pretty rudely shattered by their ruthless defeat; the better off classes might well defer to even more sweeping and effective Tsarist central rule as a condition of their own safety, and abandon all pretensions to constitutional parliamentarianism themselves, observing how it throws open the door to mob radicalism.
In short I don't think there was a soft smooth path toward western style parliamentarianism in Russia; it was a choice of authoritarianisms. Perhaps in retrospect and considering all the possibilities Bolshevism was improbable, but in that case I'd say the probable path for Russia was ongoing rightist authoritarianism.
Now note how in all of this, the health or illness of the Tsarvich has just about no bearing whatsoever on the outcomes! Given a strong institution of the all-powerful Tsar, the fact that one particular individual who inherits that power might be sickly would seem to have tremendous consequence--but in fact what would happen, as had happened frequently before in the Romanov dynasty, would be the rise of someone else as regent, or in an extreme case the usurpation of Tsarist power to a strong, healthy individual. The institution of Tsarist power rolls on majestically.
Vice versa, if somehow or other a revolutionary crisis as severe as 1917 OTL, in which the Tsarist faction has all the oxygen sucked away from it and some radical faction has power in its grasp, it is irrelevant whether the heir to the throne is in fantastic health or not. What might matter is that individual's ability to play a sharp political role, to barnstorm and connive on their own behalf to strengthen the Tsarist coalition.
In my judgement, the pro-Tsarist faction was as unified as it could be in the form of the so-called "Provisional Government" of OTL. These worthies did want to maintain the monarchy. They were divided on the issue of whether to try to keep the current Tsar, pass the succession on to the Tsar's own son, or go sideways in the Romanov succession--but I don't think it would have mattered in the least if they all rallied around one candidate or not, or if that candidate were a very clever and vigorous politician. The masses would have none of it and if anything the PG uniting behind a strong candidate would only have undermined their position relative to the Petrograd masses all the earlier.
So--of all the factors to consider, I'd say less hemophilia in the crowned heads of Europe is about the most irrelevant. It is the nature of modern monarchies, even nominally absolutist ones, that they are institutions, not expressions of personal power, and that if they are viable at all the machinery closes ranks around their sickly symbolic focus, whereas if they are not viable it does not matter if the monarch is superhuman in fact.
We can and should be looking first of all to the effect it has on the
British system not to have any heirs, even female, and for all the heirs to the throne to be foreign. And we can expect many of these effects to be initially muted, because people are going to give Victoria a good long time to start popping out heirs--a few years at least, maybe half a decade or more before they start getting very uneasy.
Victoria, if she has the personality of OTL, will not much mind not being knocked up personally, from what I understand, and she will strongly resent any effort to separate her from Albert.
The kingdom will become increasingly uneasy as the situation progresses of course. It shouldn't have a tremendous effect on the priorities of British domestic and foreign policy at first--but the absence of a certain, young and British heir will be increasingly troubling, as will the absence of that prince's also non-existent siblings to marry off in the course of dynastic diplomacy. Meanwhile the actual line of succession going through Continental, specifically German, princes and kings will indeed compel British foreign policy to cover their bets by keeping that heir in a comfortable and respectable position, which I suppose will compel intervention on behalf of Hanover. This will in turn set Britain at odds with Prussia, but may also give the other northwest German principalities a leg to stand on if they decide to form some kind of defensive federation with Hanover, thus drawing Britain in even deeper.
I'd think that by 1848 the perturbations would still be relatively minor and so the revolutionary crisis would have broadly similar effects, putting Napoleon III in charge in France for instance. But perhaps the crisis, combined with later evolutions of Chartism, would have a greater tendency to spread to Britain. After all, if the monarchy is failing to perpetuate itself on British soil and the institution puts Britain a heartbeat away from a foreign reign, in this age of rising nationalism might that not tip the balance more strongly to the republican plank the radicals put forth? Assuming that conservatism prevails in Britain, it might have to put up a nasty fight to do so, with Victoria being coopted, perhaps not unwillingly, to an increasingly reactionary position.
If Albert dies on schedule, I believe she will be under greater pressure to marry again, in the hope it might not be too late in her life for her to bear an heir at last. The OP assumes this would be fruitless since it is Victoria herself who is sterile, but no one will know that for sure. Anyway it won't work.
By the time 1901 comes round, Europe may be substantially different, mainly because of outcomes of different British policy decisions and greater anxiety in the body politic of that nation, which might cascade to produce, I would fear, a much more reactionary Britain. In fact I'd expect Victoria's reign to end quite early due to revolution--a revolution that might in the end lead to a continuation of the constitutional monarchy but with a different dynasty installed, to forestal the prospect of a German takeover. But meanwhile to cover the bet of such a German succession Britain will have been drawn into northwest German politics and come to blows with Prussia. What if the Prussians can conquer Hanover by main force despite stout British support, and drive the Welf dynasty into exile? That might solve Britain's problems in a way, but it would be bitter and liable to sow seeds of mischief. Conversely what if the British allied coalition can tip the balance back and break Prussian pretensions, so that the contenders for German unification are the Hapsburgs, the Saxon dynasty and the Hanoverian-led northwest? Will Germany remain split into these three pieces, the Catholic south and Protestant north remaining separate (but of course the Rhineland, geographically far north and probably coming under northern rule eventually, is a Catholic salient)?
Just confining our attention to probable British policy then, and its consequences, it seems likely that even in a broad strokes and high overview sort of sense, Europe in the early 20th century will look quite different, and probable alliance alignments would be substantially different from the OTL Great War polarization. The British succession will be a great crisis in itself. Getting past it--assuming for instance that some Hanoverian has had the way prepared, perhaps by being resident in Britain for decades--British ties, and perhaps the very nature of British society, will be different. Germany may not exist as such; France may still be under a Napoleonic dynasty. Perhaps liberalism will be weaker all across the board in Europe due to anxiety leading to reactionary crackdowns in its OTL womb and bastion of Britain and to an extent the Low Countries, while France does not enter a Third Republic period due to the absence of a strong German power.
Perhaps then there is no Great War crisis at all. I would guess if the political balance is shifted toward reaction economic and technical development will suffer a bit, leaving Europe backward somewhat relative to OTL, but also that radical leftist revolution will boil just under the surface. Instead of a Great War on the OTl model, perhaps there is a second '48 and the established regimes of Europe are overwhelmingly preoccupied in a grand coalition to try and suppress it.
