We know from the films that we see a clear advancement in SkyNet technology as Judgement Day gets postponed. In T1, when Sarah Connor crushes the first T800 in the hydraulic press, both cells should have been damaged, and there should have been a catastrophic failure. This obviously did not happen. Because Judgement Day was not postponed by this event, technology available to SkyNet did not improve for T2 in the T800.
In the second film, SkyNet knew that that T800 was not enough to stop Sarah Connor, so it sent an advance model back, the T1000. Since the T800 came from the same future as the first Terminator, it's power cells were also not prone to the catastrophic failure. Additionally, the power supply may not have even taken a direct hit. We see from his HUD that he is re-routing power. The primary circuits may have been cut, and the back up circuits needed to be activated. There could also be backup capacitors which would require charging time before the power relay would be completed.
At the end of T2, Judgement day is postponed, and humanity continues to innovate. New technology is available for when SkyNet become self aware in T3. This new technology included much better processing power in modern computer systems, as well as an exponential increase in processing nodes on the internet between the original 1997 J-Day and the 2004 J-day. SkyNet had a better start in the altered future, and could therefore dedicate more resources to developing better technology earlier on in its existence, possibly altering the design of the T800 to include previously unavailable technologies.
This bit is tricky, so please bear with me....
In T3, when the Terminator was explaining the damage to John Connor, he explains that he is powered by two hydrogen fuel cells. This Terminator has two power supplies, so if you allow a little flexibility with the altering of the timelines, there is always the possibility that the T1000 comes from a future where the T800 has two fuel cells, but the T2 Terminator comes from a future where it has only one high capacity battery. If the T1000 knew where to make a direct hit on a T800 with two fuel cells, it is plausible that a design with only one battery would place the power supply elsewhere, rendering such an attack ineffective.