I think that the referenced paper has all you might want (it is obviously related to the Derevnyn-Mednykh paper, but is considerably more extensive).
They do not specifically talk about orthoschemes, but their results specialize nicely.

A. D. Mednykh and  M. G. Pashkevich,
"[Elementary formulas for a hyperbolic tetrahedron](https://doi.org/10.1007/s11202-006-0079-5)",
*Sib. Math. J.*
47
(2006),
687-695.