Here's a list of data we do or don't have for American and British marksmanship:
1) Training statistics.
We do NOT have Union training statistics, but we do for the British - they used targets up to 900 yards away, and hit half the time (for First Class shots, which in some regiments was well over half the men).
2) Training amount.
The British fired about a hundred rounds per man per year, well-recorded, in marksmanship training involving range estimation and skirmishing. The Union, so far as I can tell, did not.
3) Battlefield values.
The British at the Alma, and at Inkerman, hit roughly with one round in sixteen. We do not have precise values for the Union but one round in sixteen at Pickett's Charge would imply the charge would be wiped out in three minutes and it manifestly was not.
4) Known firing ranges and achievements.
The British in the Crimea were able to snipe out artillery at 600-800 yards. This is with line infantry, not sharpshooters.
The Union was not able to get this performance from line infantry - or if they were, I'd love to see positive evidence on that front.
ED:
This is basically a structural thing, it's something that grows out of the way the US Army was organized. The US didn't have a large standing army and instead relied on building armies in a tearing hurry out of mobilized militia, but they never really managed to work up the will to enforce the training of the militia and as a consequence they were starting pretty much from scratch.
By contrast the British had a small standing army by European standards, but this functionally meant ten to fifteen times the size of the peacetime US army. As such any British force fighting in North America is going to contain tens of thousands of trained prewar regulars as stiffeners (to the extent that having one regular battalion per brigade could lead to an army nearly 200,000 strong) and going to be a much harder target than the British army.
As an example. During the potential period of a Pig War, the US Army has a few companies within a week or so of a rail line, with everything else pretty much being either fighting Indians or (out of position after) dealing with the Utah War. The British had 16 cavalry regiments, all the Guards (of which 3 battalions could leave) and 26 regular infantry battalions at home, with 4 in Canada; another ten battalions come home from India over the next year or two, and this could be accelerated at need.
So functionally the British can put together as many as five British pattern two-brigade divisions from what they have at home, or can uprate this to three corps by mixing in half a dozen volunteer or active militia battalions from Canada (plus the RCR).
Here's the positions of the US at the time:
Infantry Units
1st US Infantry: Texas
2nd: Missouri
3rd: New Mexico
4th: California
5th: Utah Expeditionary Force
6th: California
7th: Utah Expeditionary Force
8th Texas
9th: Washington Terr. and Oregon
10th: Utah Expeditionary Force
1st US Artillery (as infantry regiment): Texas
2nd: half in Kansas, half scattered along the coast as the garrisons of Ft Monroe
3rd: Oregon (in fact the infantry battalion of the 3rd US Artillery was Pickett's main force)
4th: Dakota Terr. and Minnesota
Artillery with Guns (all 4 gun batteries)
Bty I, 1st US: Utah Expeditionary Force
Bty A, 2nd US: Utah Expeditionary Force
Bty M, 2nd US: Utah Expeditionary Force
Bty C, 3rd US: Utah Expeditionary Force
Mounted Units
Coy B, 4th US Artillery: Indian Terr.
1st US Dragoons: Oregon and Wash. Terr.
2nd US Dragoons: Utah Expeditionary Force
US Mounted Rifles: Utah Expeditionary Force
1st US Cavalry: Utah Expeditionary Force
2nd US Cavalry: Utah Expeditionary Force
Functionally what this means is that the US would need to reassemble their forces at Fort Leavenworth (Kansas) and then head east. The troops in the Utah Expeditionary Force are essentially all at Camp Floyd, which is very remote and which would take months - OTL they left Leavenworth in July 1857 and didn't reach Utah until September, with Camp Floyd being deliberately selected as remote.
Meanwhile, the British would have their first reinforcements feet-dry in Canada within a month of the news steamers leaving for Britain, and troops then arriving at very roughly a battalion every 2-3 days.