How many rooms are you looking for? At £500/week, you should be able to find a 2-3 bedroom flat in N/NW London without too much difficulty, although some of the especially posh streets may be tougher and the quality/size is inversely proportional to the area/# of rooms. My wife and I just spent the past six weeks or so scouring around Bayswater, Islington, Hampstead Village, and Notting Hill (some of the most desirable areas within central London), and you can actually find some pretty amazing properties in the £500 range (@ 2bds)... and not entirely unreasonable ones @ 3bds. We really liked the Angel district, which is very hip and vibrant, but perhaps not ideal for children; some of the neighbouring areas, however, would be fantastic: Highbury Islington and Canonbury, especially, I think would be good.
I suspect one of the deeper problems with your search, though, may be that you're still in a Midwestern mindset. Quite simply, unless you are ridiculously wealthy, you will not find a flat within central London that's even a tiny fraction of what you're used to back home. Living spaces here are substantially smaller and people are forced to adjust accordingly. Within central London, for example, I don't know any family where each of their children get their own room, unless they're either rich or bought the flat decades ago. Practically speaking, at £500/week and in the more desirable central London neighbourhoods, you're likely going to be looking at places with around 800-1000 sqft. Our 2bd flat, at a similar price range, is almost 900 sqft and people say it's grand and posh.
With all of that said, if you're willing to move outwards a bit, the deals improve substantially. As a point of example, one of our fallback neighbourhoods, when we were equivocating over whether we wanted more space or a more vibrant location, was Chiswick. It's out on the edge of zone 2/3 (the areas we were looking at), but appeared to be a quite nice, wonderfully family-friendly area. In a similar price range, we were coming across large (by UK standards) houses that could easily shelter entire broods. You'll still likely find them quaint, but the adjustment may be easier to make than being crammed into an inner city flat. =) I'm sure there are comparably nice, or nicer, neighbourhoods elsewhere, where you'll get much more for your money.
Ultimately, it all boils down to desirability and money. Everyone wants to live in zone 1, right near a tube stop. Unfortunately, there is a ridiculous amount of money in this city and there's absolutely no shortage of people willing to pay ridiculous amounts of money to live in these desirable locations. My wife and I make a bit over £100k/year, and we've had to make gross compromises on flat size, neighbourhood, amenities, proximity to transportation, etc. just to find a marginally acceptable flat. And we don't even have kids yet, with all of the extra constraints they add. In the US, our salaries would be silly and we could live in the best flats, in the best locations, in the best cities, without hardly having to compromise at all. Here, when we talk to estate agents for the neighbourhoods we're (and you're) looking at, our incomes are decidedly average amongst the 2 bd seekers. It's fiercely competitive...
Anyway, I'm sort of prattling on a bit. =) Some more practical points:
- Monthly rent is calculated as such: weekly rent * 52 / 12
- The rule of thumb I've seen here is that your rent shouldn't be much more than 1/3rd of your takehome salary. Much more than that and it'll be harder to get approved.
- Don't worry about proximity to a grocery store. The practicalities of shopping here aren't at all like the Midwest and this would ridiculously constrain your options. There really aren't even that many big grocery stores here. Fortunately, most of the brands offer delivery service. IIRC, Waitrose (which is probably the poshest grocer here) has an offer where you pay something like £40/year for unlimited deliveries on orders over £50 (I'm likely mixing up some of the numbers, but the ranges are about right). Of all the things to compromise on, this should probably be one of the first. Plus, there's really not much of a shortage of local/small groceries to pick up bread, milk, etc. at.
- In terms of descending value, look for properties on: www.zoopla.co.uk, www.findaproperty.co.uk, www.globrix.com, www.gumtree.com (tons of fake ads), and then individual estate agent sites. Foxtons has the best site, and often some of the best properties, but they're a bit shady and have far-and-away the highest commissions and charges. Everyone hates them. But, they have a great site. (If you see a property you like, search for it elsewhere; landlords often multilist and, when they do, the price on non-Foxtons sites are almost always cheaper. One place we saw was a full £75/week less.)
- One of the hardest things to get used to here is that rents ARE ALWAYS NEGOTIABLE. And often by quite a lot. If it's listed through Foxtons, this is especially true. We've negotiated down by upwards of £50-75/week and had offers accepted. If you're willing to sign a longer lease (e.g. 2 years), you'll gain more flexibility.
