So I've been thinking about this question for the last day or so, and it's a tricky one for a number of reasons.

First, most of the crew on the Enola Gay did not know what their payload was. Some had suspicions it was something unusual; they knew they were training to drop some very large, unusual weapon on the enemy. But they weren't briefed on how it worked before they were in the air. And once in the air, there is no way they would have been allowed to desert their station without massive consequences; every person on an active-combat B-29 is essential.

But let's instead imagine we are talking about Bocks Car, who is about to try to repeat the same operation a few days later against Kokura (and ended up bombing Nagasaki, but that's a longer story). They knew what they were doing at that point, because Hiroshima opened up the discussion of the mission. Could they, if they had chosen to, have avoided getting on that plane legally?

My suspicion is no, but this is where it gets beyond my main knowledge, because I don't know how a few things were handled exactly during World War II. Specifically, a soldier today is obligated to disobey orders they consider to be unlawful — which, under current understanding of the laws of war, would include the indiscriminate, mass slaughter of unarmed combatants, which was without any question what the atomic bombings involved. However there are two issues here. One is that this sense of legality is a post-WWII concern; it comes out of the postwar Geneva Conventions, and was plainly not the standard that any party was acting under during the war. Hence the atomic bombings were not likely "illegal" by the standard of their day, but would probably be by the standards of today. Second, it's not clear to me how refusal of orders worked in World War II in general; a lot of our modern ethics and practice about this came out of the Nuremberg Trials, when the defense of "I was just following orders" was soundly rejected. Even then, a lot of the modern understanding of this in the US military comes from post-Vietnam, notably in reaction to the My Lai massacre in 1968.

So my hunch is that this would be seen as insubordination at the least, and desertion at the most, both of which would be serious military crimes and lead to a court martial without a doubt.

You specifically asked about conscientious objection, and that is an entirely different category. My understanding is that once you get to the point where you are actively in the war you cannot just turn around and say you are a conscientious objector. Conscientious objectors are people who refuse to take part in the violent aspects of war on principle; you don't get to be one for a specific operation or task. There were thousands of American conscientious objectors during World War II, but they were registered under a different system before they entered into the military. Someone trying to invoke this, say, the day before the Nagasaki bombing, would have probably been subject to court martial for desertion or insubordination, and would not be able to demonstrate that they had a long-standing pacifism for religious reasons.

Separately, it needs to be stressed that this was just not an issue, hence the hypothetical aspect of it. The people chosen for these planes were already committed to bombing the Japanese in one form or another; they were not being forced to do this against their will; and they were not unique in participating in the mass slaughter of the Japanese at this point (the firebombing raids had been going on since March 1945). A more interesting, general question is whether people could have objected to participating in the mass bombings against cities earlier on, and I don't know the answer to that. There were certainly members of the Army Air Forces who had moral qualms with bombing cities, even though the sentiment against the Japanese was extremely hostile (for both good reasons — the Japanese were committing atrocities, had started the war, etc. — and bad — racism). So it would not be surprising to find some who tried to get out of those missions. But I don't know of any; it is a topic I'd be interested in learning more about, because it does reflect back on modern questions about the ability of soldiers to refuse orders.