Well, Edgware Road (near Marble Arch) is Arab central - so you're likely to find a jeweller or antique dealer in that area who'll be willing to give you advice at the very least.
Alternatively, the Angel area of Islington has a really cool antiques market on Camden Passage. There are lots of quirky independent vendors around there too, so perhaps they'd be willing to purchase your antique jewellery.
Another idea would be to use the Yell search engine to locate antiques dealers in an area of London with a significant Middle Eastern population.
Knightsbridge might be a good area to start with because it's very affluent, with many Sheiks and wealth arabs frequenting the area.
Here's a search I ran just now, might be of help: http://www.yell.com/s/antiques-knightsbridge-central+london.html
Searching online means you can contact them over the phone before deciding to trek across London to see if they're interested.