I'm not answering the question, but could you provide a source for the claim that Japanese steel was one of the highest quality in the medieval world? From my reading (only an amateur), I understand that Japanese steel was actually pretty poor quality because of the low Carbon content of their iron. The Tamahagane used in making katana was made by repeatedly folding the steel to make it harder. And even so, they were brittle. And the technique was only created after 1274, so for most of the Middle Ages, they couldn't make that claim.

There are no proper sources that compare Japanese steel to any of their European (or otherwise) counterparts, let alone steel from a different time period. It is important to understand that while we understand the processes (more or less) of creating said steel, we have no way of properly comparing them as we don't know exactly how they were made at the time. There is no rigorous way that we can construct an exact replica of a sword or other weapon forged from the steel of the time, that we can test against another to say for sure. That isn't to say that we can't draw inferences or relatively well-educated guesses, but there isn't really a way to compare them.

Echoing /u/Anon4comment, medieval Japanese steel was considered 'high quality' largely because they managed to make anything at all that could work decently. The sources of iron in Japan were largely iron sand deposits - there weren't any mines like those in Europe. For this reason, the method of folding steel over into layers was more or less necessary - otherwise, you'd make a cheap pole which would bend if you looked at it the wrong way. The resultant swords however weren't necessarily brittle, but it's also important to note that swords weren't really used in battle/warfare for most of pre-modern Japanese history.

Was Japanese steel considered high quality? Depends on how you define it, but in the general sense, no - it was high quality only if you compare it to what you could have made from the same metal. Would it have been inferior to European contemporaries? Likely, but again we can't say for sure. The re-enactments and comparisons that you'll find often use replica blades or swords forged during the Tokugawa period (a long, mostly unbroken period of peace where swords were forged more for the art of swordsmandship rather than to kill someone else), rather than swords which would have more likely (not by much) been used in combat.

This answer doesn't address Toledo, Damascus, or Japanese steel, but rather "Seric" (Chinese), Parthian and "Noric" (Northern Italian) iron, which are all parts of the ancient ecosystem that Rome was a part of.

The problem with Roman "steel" is that Romans never figured out what it was that consistently created "steel" out of iron. They knew that iron ore from certain regions created stronger weapons and armor, but didn't understand that the addition of carbon was responsible. As such, iron mined from Noricum (Northern Italy/Modern day Austria) which was naturally high in manganese allowed the final carbon percentage to be higher than ore mined in other parts. Several Roman writers comment about the strength of Noric iron, so it was understood.

On the far geographical opposite of Rome, Han China did understand how additional carbon strengthened weaponry and developed a fairly advanced system of adding carbon cakes in the smelting process to create high quality steel. This steel was supposed to be limited to Han China, with penalties for trading it to other peoples, but Xiongnu raiding combined with the eventual black market for such goods started the flow of both steel and skilled workers from China to the west. This steel would have traveled along the Silk Road through Parthia to Rome, as well as down to India where it would be shipped to Roman Egypt. The knowledge of steel creation would also be passed to the Parthians where Raoul McLaughlin (my source in a lot of this) argues that such knowledge gave the Parthians the technological superiority to best the Romans at several key battles.

Unfortunately, it doesn't seem like Rome ever figured out the military applications of "Seric" or Parthian Steel as there aren't any records (that I know of) regarding legions utilizing this steel that was traded to them. Rather, they would use this high quality metal to create fine tools and other baubles in Alexandria which would be shipped to other parts of the empire.

tl;dr: Roman "steel" was essentially non-existent. They had some quality iron from Noricum, but they were operating at a technological disadvantage when compared to pretty much any of their neighbors to the east. I have done no research into Toledo, Damascus, or Japanese steel, but my understanding is all three are more attributed to the medieval period or later, and not concurrent with Rome.

Sources: Raoul McLaughlin, Rome and the Distant East: Trade Routes to the Ancient Lands of Arabia, India, and China. Bloomsbury Academic, 2010. Raoul McLaughlin, The Roman Empire and the Silk Routes: The Ancient World Economy and the Empires of Parthia, Central Asia, and Han China. Pen and Sword, 2016.