When flowing cups pass swiftly round With no allaying Thames, Our careless heads with roses bound, Our hearts with loyal flames; When thirsty grief in wine we steep, When healths and draughts go free—
More "When"s again! The second stanza is structured much like the first. We have six lines that set up a "when" scenario (as in, "When x, then y").
The speaker describes some type of celebratory or festive occasion where "flowing cups pass swiftly round." "Flowing cups" does not refer to some kind of hairstyle, but means something like "full cups" or, in modern parlance, "bottomless cups." Filled to the brim, if you will.
Sounds fun! What's everybody drinking? Well, it's not got any "allaying Thames" in it, so that must be good.
Here, "allaying" means "diluting"—i.e., watering down or making less strong.
The Thames is a very famous river that flows through London. Since the speaker is talking about wine, this is an elaborate metaphor of saying that he and his friends are not drinking wine that has been diluted with water.
In other words, they are drinking good, potent stuff—not that watered-down garbage.
Party time! On this occasion, the speaker and his friends ("our") have roses on their heads and "loyal flames" in their hearts. Apparently, they are drowning their sorrows in wine, and offering toasts.
To be "bound" by roses means the speaker and his friends have roses wrapped around their heads (probably in the shape of a headband).
There's some more funky syntax here. You should apply the word "bound" to the phrase "our hearts with loyal flames." In other words, it is as if the speaker had said "our hearts [are bound] with loyal flames."
"Flames" here means something like passion. "Loyal flames," then, means "loyal feelings or thoughts."
History note: The loyalty of the speaker and his friends is to the King of England, Charles I, for whose support Lovelace was tossed in jail to begin with. This is cleverly suggested in the use of the word "bound" in the previous line, since it gives us the image of a crown (a ring worn on one's head), which in turn suggests king through a literary device known as metonymy).
The next two lines here are tricky to negotiate. The speaker literally means, "When we steep our thirsty grief in wine, and when healths and draughts go free."
To "steep" means to soak, the way you might steep a teabag in hot water to make tea. Only, the speaker and his buddies are metaphorically soaking (think drowning) their grief in wine.
So, if this is a party, where is the grief coming from? Again, we have a subtle reference to the politics of the day. It seems that speaker and pals might be really upset by the way the country is being run. They're proclaiming their loyalty to an embattled king, and drowning their sorrows as a result.
What's more, they're likely toasting to the king at this party. A "health" is a toast (as in the expression, "I drink your health, sir"). "Draughts" are another word for cups, or drinks. Are they free of charge? Not likely. They "go free," meaning that they are offered up without restriction or hesitation. Again, we have another subtle contrast here to the speaker's own imprisonment.
Fishes that tipple in the deep Know no such liberty.
Ah. So now we learn what all that "When" business was about. When the speaker and his buddies throw their pro-king pity party, not even the fishes in the ocean are as free (as they are).
"Tipple" is an old word that means "drink," and "deep" is a word often used in poetry to refer to the ocean. So, in other words, fish—which are able to drink (really, breathe) all the water in the ocean—aren't enjoying as much freedom as the speaker and his friend at their drinking fest.
The point is that the speaker and his friends possess some quality (imagination or reason) that fish do not.
In the first stanza, the speaker claimed to be free, even in prison. The same type of paradox is at work here. The speaker boasts again of his seemingly unlimited freedom. In fact, the very same phrase, "Know no such liberty," is used as well. Hmm. We may have a refrain on our hands here!