Hypothetically, let’s say the statute actually had a section with “no extraditions for unsavoury offences” in it. That’s pretty weird and ambiguous, so you might, subject to UK law on statutory interpretation, look to the treaty/treaties the Act is implementing to figure out what the legislature meant by unsavoury when they wrote it. Maybe a bunch of treaties had similar provisions using the word unsavoury, but the US one changed it, and you would analyse why they didn’t use the language again, or whether political offences as referred to by the US treaty would fit the bill... But according to the judgment, the Act is not ambiguous. There is nothing to look to the treaty for.
Still, the basis for the extradition is not the treaty. Treaties bind States (ie countries) against each other. The remedy for a treaty law breach is stern words from the UN, maybe a fine, whatever. But treaty law cannot establish domestic laws that govern things like extradition. The only requirement imposed by the treaty is on the UK as a State to implement the treaty in domestic law, which is how countries like the UK comply with the terms of the treaty. International law does not bind local decision makers who decide whether the extradition goes ahead. It also does not bind parliament, which can refuse to implement a treaty or decide to deviate from it. It should also be quite plain that Assange is not a State party to the extradition agreement, being neither the literal United States nor Kingdom, so he does not have standing to object to the UK’s implementation, and of course, is in utterly the wrong court for that :)
(That the UK is bound to comply makes for a strong suggestion that parliament actually intended to be true to the treaty when they implemented it, but this is only relevant where there is ambiguity in the domestic law requiring resolution, because of the primacy of legislative power and its ability to write laws in clear terms that can’t be wriggled out of.)