Interesting question. Ideally, a king--any king--knows that he's only a servant of the people, and that he's still subject to God's laws. This implies that while he is still the ultimate civil authority in a land, he still works under the guidance of the religious leader (in Noah's case, the priests). At the very least, he's spiritually obligated to at least listen to the advice of the priests or prophet. In this ideal situation, a man might very well be tried on both a civil and a religious trial, though one would hope that the king's judgment would already be tempered by spiritual understanding and so only one trial would be needed in the first place.
However, Noah's situation was far from ideal. For one thing, the man was certainly not concerning himself with spiritual matters. He was worldly in the worst sense of the word, using his authority as king to justify some very grave sins against God. As bad as this was, however, Noah still had to contend with the priests. His father had been a good man who'd placed great value on the advice of his spiritual leaders, and no matter what Noah himself thought of God, he knew better than to do away with the priests after his father died. The people, who would become little better than Noah himself, would probably have objected or at least questioned such an action, so Noah instead chose to do something that actually gave him even more power: he got the priests on his side. He either seduced the priests away from God or simply replaced them with ones who would give him "permission" to do whatever he wanted. He found men who would claim to know God's mind and then use their supposed spiritual authority to justify any sin Noah wanted to commit. And with God and the law supposedly both backing him up, Noah convinced the people that evil was good and that committing sin was somehow obedience to God.
And all of this just means that not only would the priests not have challenged the king, but they probably encouraged him to condemn Abinadi. They wouldn't have gone against the king because the king was the one giving them their own power--God obviously had not given them any, and they probably knew that--and Noah wouldn't have argued with the priests since they were the ones providing the justification for his own sins. One trial, then, was all that was needed, because both the secular and religious authorities of the land, no matter how false both were, acted as one when they condemned Abinadi and later ordered his execution. Although of course God could have stopped it all, Abinadi never really had a chance.
...and he knew that going into this situation, because of course Abinadi had at least spent a great deal of time in the area and knew exactly what was going on in that city. He'd have known about the political factors, as well as knowing about the false priests, but he went anyway. He knew he'd be killed, and he probably knew his death would not be an easy one. Still, the gospel and his obedience to God meant more to him than his life or a painless death, which is precisely why we admire him so. He truly was a great man.
Still, isn't it ironic that one of those same priests eventually turned everything around and ultimately created the ideal situation that I'd mentioned earlier? Alma's relationship with King Mosiah was exactly what it should have been, which is probably why the people flourished so much under their joint care. Makes one wonder what the world would be like if our own leaders listened to the advice of the prophets, doesn't it?