Neutrality is not only plausible, it's pretty much mandatory. 1848 may not have produced a major rebellion in Britain, but it did wipe out most of the accumulated equity in the financial markets; then a round of bond financing to pay for the Crimea, 1851-1854; then a round of bond financing to pay for the Sepoy Uprising, 1857-58. A third round of bond financing for a significant foreign war in 1861 will be just what it takes to destroy the British bond market altogether, and the Chancellor of the Exchequer knows that, no matter who he is. The Confederates don't understand that, of course, but their rebellion is timed almost perfectly to make British support impossible.
Napoleon III's France can intervene in part because Nappy understands a good deal less about the financial consequences to himself.
What does "German Empire supports the USA" look like? To do what? The two nations have no common interests at this point, aside from perhaps a mutual dislike of France, but that isn't going to amount to anything, particularly since the likelihood of rump USA being a great naval power in the 1870s is basically zero.