Is the rate of electrolysis the same throughout as long as voltage is same In an electrolytic cell, where the electrolysis of dilute sulfuric acid is taking place, would the rate of electrolysis be constant always, even at the beginning?

By rate of electrolysis, I mean the rate of gas production or the rate of charge delivery. And would this rate be established as soon as I turn on the voltage, or does it slowly increase to a constant level?
The rate of electrolysis (decomposition of water) is defined by the current that passes. If you apply a constant voltage, factors arise as suggested in the comment by MaxW.

At constant current, the rate of hydrogen (and oxygen) evolution would be somewhat different initially because on a microscopic level, hydrogen atoms have to cover the surface to some extent to form H2; H atoms may reduce some oxides on the electrode surface, and H2 has to agglomerate to form bubbles (and dissolve in water to saturate it with H2). And then the bubbles have to rise to the surface.