What is the scaling for % completion time in a Michaelis–Menten scenario as a function of substrate concentration?
In short, yes.

To obtain the relation between τ and v, we need the actual time evolution of the concentration of the product by solving the differential equations (DEs) and then some way to get steady-state concentration. The DEs are non-linear. So we have to find approximate solutions. This paper develops a solution. It's not nice..

I will probably come back to add more. For the time being, I highly recommend those interested to browse through the paper.

And what do you mean by τ≈1 leading to failure of steady-state approximation please?

Edit: The tittle of the paper is "Analytical solution of coupled nonlinear rate equations : I. Michaelis-menten kinetics" by Phillipson