"Peasouper" is the term given to a type of smog - now, thanks to clean air regs a thing of the past - which was so thick and heavy it was like trying to walk through pea soup.
"Flying rat" is a pigeon.
"Paved with" is an allusion to the idea that London's streets are "paved with gold" but in fact are paved with... well... I think the brown explains that.
"Cuppa" is meant to be the colour of tea, but that shade is too milky for me. I am a Northerner who ended up here by accident 12 years ago and decided to stay, though.
"Pint of Pride" refer to Fuller's Pride. Fuller's is a brewery in Chiswick that makes "Pride" to this day. It is a common site in many London pubs, and the only thing better than it in most pubs if you find it is Timothy Taylor Landlord. Do not drink Green King IPA - it's terrible. Pride is bit darker than that orange though, which is closer to Iron-Bru.
"Parakeet" is often a surprise. Short version: some pet parakeets escaped (or were let loose) decades ago. They bred. They're now everywhere. You'll see wild parakeets across London often. They kind of ruin the quiet serenity of the parks, but it's not their fault, and they are pretty against a grey London sky.