not having seen the results, it's hard to tell where you may have gone wrong, and there's nothing glaringly wrong with the recipe either. for what it's worth, here are some points where you may want to pay attention to on your next try.
as far as the recipe itself goes, the one point that i'd question is the nut filling. i could be wrong, but with 1/2 c milk along with the meringue, it sounds like it could be kind of soupy. where i would head as far as the fillings go would be that they be of a consistency that is soft enough to be easily spreadable without tearing the dough or, that you can sprinkle on.
the recipe says to "Beat in remaining flour until mixture forms a soft dough" and then refigerate and use, but you should be kneading it well until the gluten is developed. by hand maybe 15 minutes???
also, the recipe calls for AP flour but bread flour (hard wheat) would give you better results as far as strength goes. AP flour without kneading would give you a comparatively cakier (and weaker) consistency.
make sure thqt the nuts are chopped or broken small enough that they won't tear holes in the rolled out dough.
when you roll up the rolls, you'll want to tread the fine line between rolling up firmly enough to keep out as much air as possible, and yet not pushing too hard and tearing holes in the layers.
and when you pinch the seams, do the entire length and making sure the the end is adhering well but, again, without tearing holes.
finally, when you transfer it a cooling rack, make sure that it's entirely supported along its length, and not just pick it up at the middle or ends only. you may also consider using enough parchment under each one that you could transfer it by pick up the parchment along the sides and "cradling" it. you could also bake them already on the cooling racks so that you could let them rest 10 minutes or so and then simply slide the parchment out from under them.
better luck on your next try, and don't give up. they sound delicious.