If you get the chance, check out Duncan Andrew Campbell’s “English Public Opinion and the American Civil War”, where he argues that English public opinion in general was toward a strict neutrality (because there was a general dislike of both sides), and that only a very small minority supported intervention. The majority of both the public and of Parliament figured they might as well just let the Americans kill each other.
I agree with him in the long run. But in the short run and in local areas, there were major differences from one side to another. And mid-1863 was one such time.
But he argues that there was never any time during the period of 1861-1865 when there was sufficient political support for English intervention/recognition…not even in 1861, during the Trent affair. There certainly wasn’t in 1863, after the Emancipation Proclamation had been issued and Egyptian cotton production was up.
Now, did that mean that there weren’t any Britons who were sympathetic to the Confederacy, or that there wasn’t a good deal of schadenfreude going on in Britain about the US’s situation? Of course not. But that’s a far cry from saying that British recognition was on the cusp of occurring.
In which case I disagree with him.
It seems to me that the South early on, as the won some improbably victories, started believing in God’s Providence that the South would win, which I think affected a lot of the deciosn-makers in the South. I think it even affected Lee to a point, potentially fatally, when he ordered Pickett’s charge at Gettysburgh. Of course while one argues what might have happened had the South won at Gettysburg, the war might have also ended right after Gettysburg had Meade pursued the tired, defeated, retreating Rebel army right after the battle.
First of all, I am not the one advocating the Lee would have done anything of the kind. I have consistently said he wouldn’t have done that at all. I think you have me mixed up with someone else. 
Second of all, as is ably argued by others here, the English were not really interested in 1863 in going over to the side of the Confederacy. But that doesn’t mean that thoughts of encouraging that decision didn’t motivate Lee.
According to Bruce Catton, who says it is based upon Lee’s own statements after the war about his purpose, the Gettysburg campaign was about accomplishing two main things: get Hooker away from Richmond, and find a way to supply his army for a little while that didn’t depend upon Virginia (Catton, Never Call Retreat, p. 159 (1965)). You will notice that this would not have entailed any attempt to go in the direction of Washington, D.C. But it also is not intended as a true attempt to gain a military victory. There were thoughts of pressuring the North into a political situation where a negotiated peace might occur, but frankly, that was unlikely to occur anytime after the Emacipation Proclamation, because the South was never, ever going to accept a peace that adjusted the relationship of the landed gentry to their “property.”
Do you base that disagreement upon any particular evidence?
I tend to agree with your assertions about a Southern victory at Gettysburg causing a blow to prestige, though frankly I suspect that the Union army would have been forced to engage him again before the end of July, and most certainly would have won a second battle, given that they would have been reinforced, whereas Lee could not have been.