I think it all goes back to timing. When do the British declare and what is the situation in the world and the war
The most obvious one is always Trent, as that's the one with the least barriers to British involvement.
The second most obvious one is basically a multilateral intervention, which would require:
1) Britain, France and Russia agreeing to it.
2) All three powers proposing mediation.
3) The Union declining and the Confederacy accepting.
In that case, definitionally Britain+France+Russia all declare war on the Union to get them to submit to mediation.
After that you're
probably down to the following.
The British officially recognize the Confederacy as a nation. (Federal DoW follows.) Unlikely but possible.
The Union in some way violates British neutrality unforgivably. (Trent mk. 2, firing on a British warship and not apologizing, conscripting British subjects into the army...) Possible IMO.
The British do something which the Union declares war on them for. (e.g. Laird Rams). Also possible - though unlikely.
Union goes full vengeance on the Confederacy, with mass executions etc. Probably nearly impossible but could provoke a British intervention from sheer horror.
The latter four could all happen any time in the war. The multilateral intervention was OTL considered for about a month (the month pre-Antietam) and never got far. Trent is Trent.