Adama didn't need proof. I am pretty sure he believed her.
Be that as it may;
They needed proof to show to the public. If they were just to come out and say *on election day* "Baltar is a Cylon conspirator" and put him in the brig, it would be just a tad suspicious to the voters. It would look like Roslin and Adama basically staged a coup on the pretense of an unqualified rumor. The civilians were already getting more and more uneasy. There would most likely be a rebellion (no doubt led by Zarek). The fleet would split, etc, etc.
And I *really* don't think Adama believed the nuke was stolen. He was reporting the "facts" to Baltar, but to me, his tone said he knew exactly how the bomb got out of his lab and into the hands of the Cylons. But he was playing the good soldier, and following his orders.
the more they should have stopped him on election day if he was a cylon conspirator rather than give him the power of the presidency. if they wanted proof they needed only to follow him because after he was already president, he still went to see cylon 6 and fracked her. how could he have moved around so easily as before and see the cylon without anyone finding out? its just stupid. as stupid as him being able to move the warhead in a briefcase. and it was so unlike adama to not do anything, he staged a coup when starbuck followed Roslin's orders but he sat on his ass knowing the president might be working with the cylons? he should have staged another coup or just let Roslin remain president. yes, there might be a rebellion and the fleet might split, but that would make for an even more interesting storyline.