Is it ever specified that whether each alternate timeline has a ToH of its own, or there is only a single ToH in the entire Type-Moon multiverse?
The Throne of Heroes exists outside of the world and the time axis. This interpretation is taken from Fate/complete material III: World material - Records of Heavens Feel - Servant System: Difference between Heroic Spirits and Servants, p.012, you can check a fan-translation from that excerpt here. So there's only ONE Throne of Heroes, which is linked to the Root which is also cut off from the world and the time axis.
There is a single Throne of Heroes for the entire multiverse (Kaleidoscope), which is why you can have different aspects of the same heroic spirit.
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