Actually, there is a reason. First, they need to read the last rights. We're not wholly sure of how worshipping the Divines works for them, but it seems that they do need to give last rites. The "Nameless Stormcloak" interruptes the rites, and basically says "Screw this, I'm first!" before marching up to the chopping block.
Second, Dragons haven't been seen for over 5,000 years, and they're deep within imperial territory at the time of the execution, in the middle of a imperial fort. Weather or not Ulfric goes first, his death is pretty much assured
under normal circumstances. Are you going to blame the navy that gets defeated by a tidal wave brought on by a freak storm for not being prepared?
In fact, Ulfric may have been first had that one Stormcloak not interrupted. And if Alduin didn't decide to attack the fort.