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(Apologies in advance: not being American, while I've read up somewhat about the military aspects of the Civil War, I am somewhat unclear as to its political dimension.)

Lincoln did not declare the slaves free until 1863, despite his personal beliefs.

Did he wait because he was not sure of the political support in the North for accepting Abolition as a war aim, along with stopping Secession? Or did he wait to preserve the possibility of negotiations with the Confederacy? If it was the second reason, is there any indication of what he was willing to compromise on? If the first, what changed?

Yes, I've seen What is the context of Lincoln saying: "if I could save the union without freeing any slave I would do it" but it doesn't tell me why he waited and what got him to commit to abolition.

@Peter Diehr's answer is a good one and I've upvoted it - but I'd like to expand on that, remembering that Lincoln was a very good strategist.

First, The Civil War was about slavery, and the South's (correct) understanding was that the North was growing faster than it was and free states would sooner or later substantially outnumber slave states. (Some of the nominally slave states allowed slavery, but were not economically dependent on it as were the Deep South states. Sentiment for abolition was growing in those states, also. Slavery was doomed under the Union.)

If the South did nothing, they lost in maybe twenty years when the North would have the political power to abolish slavery. They had to secede when they did.

Lincoln, OTOH, had history on his side. If he could hold the Union together, then slavery was doomed. Until the war was beyond settlement, his best bet to abolish slavery was to try to keep the Union together. So in the early days he insisted his only purpose was to hold the Union together.

Secondly, the Union included several border states which allowed slavery, but where it wasn't the bedrock of the economy. To have a decent chance of winning the war militarily, the North had to hold on to those states. So even the Emancipation Proclamation didn't free the slaves in Union states! Key point: For the Emancipation Proclamation to have effect anywhere it needed to not push the slave states still in the Union into rebellion. So it had to exempt them.

And then thirdly, as Peter says, once Lincoln had a strong Union victory he could start pushing explicitly on ending slavery. Once the North (and foreign countries, especially Great Britain) saw that the Union would end slavery and had to power to do it its moral authority was immeasurably strengthened which sped the end of the war.

Lincoln waited until there was a great union victory; the early losses, and the poor showing of the Union generals did not give him a very firm place to stand and make promises.

Antietam was the victory he was waiting for, and a preliminary declaration was issued at that time.

See this History channel article on the Emancipation Proclamation for further details and analysis.