Is the rate of electrolysis the same throughout as long as voltage is same
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.