Both.
The way I see it the earlier into someones un-life the faster they are. When someones immediately turns they've basically got a human body but with no pain limits, so they're capable of near-superhuman strength (like Meth addicts) and incredible speed as they can't feel any fatigue. But after a while all the damage caused by their activities, muscles torn, bones broken and other factor slows them, because they have no regenerative processes. They become the shambling undead.
That said of course the body before zombiefication plays into it as well. Fat people will of course be slower than say athletic people when they become zombies.
The way I see it the earlier into someones un-life the faster they are. When someones immediately turns they've basically got a human body but with no pain limits, so they're capable of near-superhuman strength (like Meth addicts) and incredible speed as they can't feel any fatigue. But after a while all the damage caused by their activities, muscles torn, bones broken and other factor slows them, because they have no regenerative processes. They become the shambling undead.
That said of course the body before zombiefication plays into it as well. Fat people will of course be slower than say athletic people when they become zombies.