Ex-pats from quarter century ago, and hubby born and bred Islington North London and I worked at St Barthlomew's hospital back in early 80's.
As a local, we loved going to Upper St market at weekends Islington, also another called Club Row (can't remember it's exact location). Petticoat Lane was another favourite - none of the markets are the same as yesteryears of course :(
We love walking along the Camden locks, seeing the houseboats and as some enter the locks. On a visit back there in 2011 and 2015 (first return for youngest daughter to see her history, then hubby in 2015 who hadn't been back for over 18 years), we walked from Kings cross station all along the canals to Camden Town, then caught a bus back up to Upper Street Market and walked back down to Kings Cross from their to Tube it back to Victoria.
Sadly there used to a good old fashioned British pub on every street corner, most of our old young haunts now though have been torn down and high density flats/masionettes have been built in their place. Soho, Bond Street, Regent Street, Hatton Gardens and many of the other places all looked so different, and the crowds - Holy guacamole is there ever a ton of folks with no manners - crossing to access main line tube stations was like a charge of the light brigade with no sense of one direction to one side. Quite the eye opener from our yesteryears. Sigh.
We stayed at the University accommodations both times, just around the corner (quiet treelined street) but within easy walking distance to Victoria Tube station. Get an Oyster card and load it for your weeks stay! We walked for miles and miles without using transport much at all. Buckingham Palace, Clarence House, Windsor Castle, Thames River Cruise, Tower of London, London Zoo, Tower Bridge, Westminster Abbey, Houses Of Parliament, No 10 Downing Street, Big Ben, Trafalgar Square, Piccadilly Circus, Harrods (because you can!), Madam Tussauds, Churchill War Rooms?
Maybe check out the London Pass if it will save you some money admissions. There's nothing like being in a cab with an old born and bred true Cockney - if you find one it'd be worth the entertainment not only on "lingo" (language) used but they'll talk about "the old queen mum", and tons of interesting facts. Maybe ask for a set rate, would they drive you around past the main attractions and ask "where's some hidden local secret must dos".
Decide from googling what interests you Gary, then googlemap it to create your daily itineraries. Oh a little out but Windsor Castle you might find interesting as well. We also liked Woburn Abbey further afield.
FTW.
PS: If you want I have a wonderful generations born and bred childhood friend of my husband's who's a cabbie (taxi driver) I can give you his face book contact if you PM me.
PPS: Enjoy the original British culture and ambience if you can find it, just take plenty of "dough" (money) Some Cockney Slang you might hear = Dog and Bone (phone), Apples & Pears (stairs), Frog and Toad (Road), Blue Lady (UK5pound note), Mincers and/or Mince Pies (hands and eyes) and other money slang = pony, monkey, quid. Above all embrace it and have an absolute blast.
