“Look kids, Parliament…Big Ben!”
I was born and brought up in south-east London, so:
Greenwich Park
The National Maritime Museum
Cutty Sark
Gypsy Moth (Now restored and back at sea where she belongs)
Hawksmoor’s church in West Greenwich
The Royal Observatory
Lewisham Market
Woolwich Tram Shed
The Plume of Feathers pub (also The Yacht, The Cutty Sark, The Trafalgar Tavern and the Anchor and Hope).
The east terrace at The Valley. Like watching football from the north face of the Eiger.
The foot tunnels at Greenwich and Woolwich
Woolwich Ferry
The Docks (when they still were docks)
Vanborough Castle
Blackheath
Blackheath Village and the boating pond…
I visited London, England, for the first time when I was attending for a job interview a few years ago.
My first impressions as a wide-eyed country lad and what sticks in my mind are:
The graffitti, dirt and grime and how run-down it was, most especially in the train station where I first arrived (I was really surprised at that)
Sitting opposite someone who was wearing a racist badge while on the tube, I can’t recall the exact wording but it was basically telling immigrants to get out of England. I wondered what reaction I would get if I engaged him in conversation, I’m white but not English and this was in the mid-90’s when the IRA was still running a terrorist campaign in London.
How expensive everything was, when I bought a can of coke I thought there must have been some mistake.
The crowds and rush of activity, the different accents and cultures evident everywhere.
On a somewhat amusing note when talking to a Pakistani gentleman he asked me what part of Australia I was from while a Chinese lady asked where in America I was from, my Irish (more precisely Ulster) accent seemed to throw them but at least they seemed to have no trouble understanding me, unlike some of the English people, though I strongly suspect they were just being thran. 
That may seem overly negative which wasn’t my intention, I enjoyed my time there and have been back several times since.
However I’m not really a big city person and the OP did ask what pops into your mind when you think of the place.
I actually got to do this last autumn when we were driving up there to see Brian Wilson; we ended up driving around Parliament Square three times having missed a turn. My English companion, who was driving, was not amused.
What I think of when I think of London – that I have to remember to reserve a manuscript at the British Library for when I’m up there in May.
There’s a lot of other associations and thoughts, but ‘home’ is one, too, even though my stints living there were all too short. Still, after September, ‘home’ will just be a short train ride away from the Smoke! Yay!
Home
They don’t, it’s just a reflex. It took this American a couple of months to train myself to look in both directions. I never could get myself to look to the right only. (Five years later in America I still look in both directions before stepping into the street, every time. Probably a good habit anyway.)