The typical English-influenced way to do it starts with the number,the most specific thing.

  10 Downing Street
  London
  SW1A 2AA

(The postcode is least-specific first: SW is South West London, 1 is the district nearest Central London, and the 2 would usually be a few streets and the AA 5-10 houses, but this is a unique code for an address that gets a lot of post.)

Funnily enough, with the addition of the newest element to that format — the postcode — to the end, this format also becomes less "logical", postcode being very specific.

Of course that's arguable, as postcode shouldn't really be considered a component of-, rather an alternative to- other sections of the address.

A postcode (or local equivalent) isn't necessarily particularly specific. For example, here in Austria the equivalent of a postcode might cover a whole city district.

Yes, similar to the US. But British post codes (or at least the ones I’ve seen in London) are in fact very specific.

Indeed, a single postcode in the UK will contain (on average) about 20 addresses and (usually) only refer to a single street.

ZIP+4 in the US is quite specific, though. E.g., my ZIP code is probably > 50k people but the ZIP+4 is just my apartment building.