This is my European perspective on the question:
The safest way to get the Right to distance itself from Nationalism is for the Left to cling to Nationalism.
This requires an egalitarian interpretation of Nationalism, and a lot more "democracy" and "proto-socialism" and less "liberalism" in Nationalism.
It also requires no Marxism, but since such a turn of events would probably require a different course of the French Revolution, Marxism is likely to be butterflied anyway.
So, with an egalitarian, democratic, populist, Nationalist Left (there's always space for a few deviations, like non-nationalist anarchists within the Left, or something akin to OTL's internationalist socialists, but they'd have to be fringe voices, like anarchists and left-wing nationalists have been for much of Europe's 20th and 21st century), I suppose it makes sense to assume that the Right is monarchist, anti-secularist Christian, economically much more radically pro-capitalist, defending established privileges (among them bourgeois liberties, too, over time this becomes the main interpretation of this tenet while at first it also refers to aristocratic privileges and the various special laws of towns, church and monastery holdings etc.), emphasising property and traditional legal order, fighting fiercely against universal suffrage and maintaining elitist and anti-populist positions throughout the 20th century. Right-wing anti-Nationalism would spring from such diverse sources as Catholicism (and Christianity in general, and over the course of the 19th century increasingly also from Judaism), bourgeois cosmopolitanism, the aristocracy's traditionally close interrelations across Europe, and probably also scientific universalism, if the latter and the religious nature can co-exist peacefully, which I believe they could. The Right would continue to emphasise and support the social peace and stability brought about by not tampering with the established (pre-national) social order, hierarchies (arching over national boundaries and certainly not making all members of one ethno-linguistic group "equal"), differences between individuals, particular interests etc. and denounce Nationalism as a strategy of populists trying to manipulate the uneducated proles for their own goals.