Specifically in the news how where the facts of the war, the scale, individual battles covered? Would, for example, The Economist, report on the Battle of Antietam? Would it be compared to European warfare? Was the significance of battles and campaigns understood?
Google has a newspaper archive with digitized copies of old newspapers, some of them going back to the 19th century. A good one to look at is the Glasgow Herald, which they have an extensive archive of.
If you look at the editions from the 1860's they cover the American war in a fair amount of depth. 19th century newspapers tend to have advertisements on the front page, and the news coverage usually starts on page 3 or so.
What strikes you when looking at the Herald is that the news from America is usually a couple of weeks out of date, as you would expect with shipping times. You mention Antietam, so here's the edition from October 1st, 1862, which covers it. https://news.google.com/newspapers?nid=GGgVawPscysC&dat=18621001&printsec=frontpage&hl=en
"Mrs. Lincoln accused of treason". Then in the next column, there's news of a battle, which I think is Antietam. "New York, Sept. 20 (Morning). Later accounts from McClellan's head-quarters state that Tuesday last was passed chiefly in deploying forces and gaining positions, but a severe battle was fought on Wednesday, lasting from dawn until dusk.
The result of the battle was indecisive. The superiority of the position remained with the Federals. The Federal loss was estimated at 6000 to 10,000. The Federal General Mansfield was killed, and Generals Hooker, Duryea, Summer, Meagher, Max, Weber, Dana, Hartsfuff, Richardson, Sedgwick, French, Ricketts, and Rodman were wounded. The loss of the Federal Generals and field officers is stated to be so large as to be unaccountable. Little but skirmishing occurred on Thursday. General McClellan officially reports on Friday morning that the enemy had abandoned his positions on Thursday night, leaving the dead and wounded on the field; and says, I do not know if the enemy is falling back to an interior position, or crossing the river. We may safely claim the victory as ours. Later, he reports General Pleasanton is driving the enemy across the river. Our victory was complete. The enemy is driven back into Virginia. Maryland and Pennsylvania are now safe.
The newspaper accounts say the Confederates commenced crossing Shepherdstown Bridge, and two fords above and below it, early on Thursday night. During the night McClellan advanced a battery, and shelled them from the surrounding heights. Early on Friday morning General Pleasanton pursued the Confederates with two batteries and two regiments of infantry, and captured some of their ammunition and supplies, and a small portion of a South Carolina brigade. "Stonewall" Jackson conducted the retreat into Virginia, and the whole Confederate army appears to have crossed the Potomac with but alight loss of men, waggons, or artillery. The Confederates retreated in the direction of Winchester. A considerable Federal force is on the Virginian side of the Potomac, and speculation is rife as to whether this force will interfere with the Confederate retreat.
It is not decidedly known whether the Federals or Confederates occupy Harper's Ferry, but it is rumoured that Burnside yesterday crossed into Virginia via Harper's Ferry, and was moving against the Confederates.
McClellan's army had crossed Antietam Creek, and is massed between Antiatem Creek and the Potomac, opposite Shepherdstown. It was supposed that McClellan would cross the Potomac." etc.
There follows more in the same vein, a financial report on the state of the market in America, and the text of "General Lee's Proclamation".
I assume they were just getting this information from New York newspapers, received a couple of weeks after publication. Someone else will know if they had established transatlantic telegraph communication at this point.
If you're interested in how the Confederate government tried creating a positive image of itself in Great Britain as well as a negative image of the north, check out The Index. Henry Hotze was a Confederate agent in London and he created a newspaper dedicated to promoting the Confederacy and influencing public opinion.
You can view some of it here https://archive.org/details/indexweeklyjourn02hotz
Also, Cause of All Nations is a very good book that touches on this subject a LOT!