Obsessive googling has turned up the name of A.Gold in Spitalfields as being a purveyor of classic British baked goods like Banbury tarts as well as no end of cheese and chutneys and old fashioned foods

http://www.agoldshop.com/index.html

http://www.productsifter.com/Places/...ld-review.aspx

I went to the Patiessrie Valerie about a block from the Ritz. In addition to pastries, they also sell ice cream. I believe there is also one near Covent Garden, in direction of Trafalgar Sq.

I am a great fan or A.Gold.
The shop is well worth a visit.

Aduchamp1 - You made me laugh at the thought of using a bagel for a target! Clay Pigeon shooting is great fun and the last time my DH scored a 'hit' was from the back of a cruise ship.
Thanks for the laugh but I think most missed it!

Not English cakes per se but you might like a chain called Ambaala which is an Indian chain selling traditional indian Sweets.

Also look out for Turkish places.

Not in the centre - but common elsewhere.