Is there only a single Throne of Heroes in the entire Type-Moon multiverse? 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.