Oh yay, my city (well, with Tel-Aviv). Let me try and wrack my brains...
Firstly, if the weather's good, you have to head up to Hampstead in North London - my beloved home. England's capital during the great plague, it was swallowed up by London but maintains its hill-station feel remaining a village amidst the sprawl, and home to intellectuals, artists and authors both living and dead. Crucially, it abutts Hampstead Heath, an inner-city forrest with swimming ponds, lush vegetation and the beautiful Kenwood House. Further afield is the village of Highgate with the Highgate cemetery - a gothic structure where Karl Marx, among others, is buried.
Further south is the West End (all of W1), a mish-mash of neighbourhoods that pretty much sums up London as a collective mess. Soho is bohemian and arty: no chain-stores, whore-houses (sorry, "Models 1st Floor"), milliners and post-production studios rub shoulders with the centre of the London gay scene; across Shaftesbury Avenue you'll find China Town and Covent Garden (theatre-town and 'hippy' stores); cross Lower Regent's St. and you're in St. James - the home of the true English (a dying breed in London), 500-year old stores and the royal households; cross Picadilly and you're in Mayfair - swish stores, ultra-trendy restaurants and the new moneyed elite; cross Oxford St. and you're in Marylebone - doctors, Jews and Arabs as well as one of the nicest High Streets in the city (Marylebone High St.) and the wonderful and bizarre Wallace Collection; cross Great Portland St. and you're in Fitzrovia - a lovely, odd neighbourhood and home of the BT Tower, London's discount electronics street (Tottenham Court Road) and Charlotte Street: my favourite collection of restaurants, pubs and bars (Koba for Korean on Rathbone Place is a personal favourite); cross further to the other side of Tottenham Court Road and you're in University Land (Bloomsbury) - students and nutty professors are the order of the day, many of the latter owning run-down book stores themselves in a haze of stale cigarette smoke, and home to the British Museum. There's also the wonderful R G Lewis for cameras (well, technically it's in Holborn, but it's within the margin of error).
All walkable in a day, and well worth it to get a feel for the centre of London.
As said above, the South Bank is great (go to the Tate Britain and then head over the river at Westminster Bridge) - walk east as far as your little legs will take you.
East End - Brick Lane's a bit of a nightmare. If you're not getting solicited by Bangladeshi curry houses, you'll be trying to avoid the upper-middle-class Hipsters that have colonised the area. For creative neighbourhoods, everything's moved to Shoreditch, Hoxton and Dalston - small contemporary galleries and stores. If you do go to Brick Lane, head around the corner to Tayyabs - the only place to get a curry, a local institution, dirt cheap and the best curry I've ever had - on Fieldgate St.
I'd also go to the Docklands - just to get an idea of where the city's headed. It's a painful, scary, empty soulless place, but interesting just for the clinical inhumanity of it.
As for photographic stuff: I've never had a problem photographing in London. I've taken photos in The City, The London Eye, the underground, The Docklands, everywhere. Just don't rock up with a tripod and you should be ok.
Specific photographic places: South Bank, South Bank centre (a brutalist maze of concrete that everyone in London is very fond of), Liberty's department store (a mock-tudor shopping emporium - and yes, I've taken photos there plenty of times), Parliament Hill (on Hampstead Heath) and/or Primrose Hill for views of the London skyline.
Hope that helps... London's a huge, sprawling mess and is so diverse that it's difficult to get to know or get your head around. But then again, that's why we love it.
PS: If you want any info about food, I think you should ask. The food in London can be either god-awful or amazing, and unfortunately, the ratio is about 90:10. I'd be happy to help, or otherwise I fear you'll end up in Angus Steak House after Pizza Express after Garfunkels....
