Oh boy, my favourite topic!
I have a few suggestions:
Robert Dyas: household goods, cleaning supplies, electrical bits (kettles, toasters, fans etc).
http://www.robertdyas.co.ukArgos: an experience in itself. Argos.co.uk
Marks & Spencer: some pretty good deals on bedding, pillows etc.
http://www.marksandspencer.comSelby's: this dated department store in N7 on Holloway Road has great deals on bedding/duvets/pillows.
John Lewis: Probably the closest to a Macy's; you can get absolutely everything here, pretty decent looking, if not sometimes pricey, furniture. All major appliances etc.
http://www.johnlewis.comWhite Company: ££ on the linen side, but if you are looking for classy stuff, here's where you'll find it.
http://www.thewhitecompany.comPoundland is a great place - don't knock it! Also, you could do worse than get an Amazon prime membership; they ship within one day and the delivery windows are usually 2 hours. Really slick operation over here. Oh, and check your local high street for an "anything/everything" shop. They look like they could be hardware stores but when you go in they literally have absolutely everything. And cheap.
Achica.com - one of those flash sale sites. Be careful: their delivery dates may be ages.
Made.com - if you don't mind waiting 10-14 weeks for your sofa.
Gumtree.com - go here for furniture finds and anything you might have searched craigslist for (including free stuff). Problem is you have to pick stuff up!
Freecycle - alive and well in London.
Zipcar: look into it. £9 van rentals to go pick up that cheap, vintage chest of drawers you find on Gumtree/Ebay/Freecycle/Ikea (<-- don't forget about them!)
Mugs: check out Emma Bridgewater. Her Toast range is awesome. There's an outlet store at Bicester Village (Oxfordshire - regular trains there from Marylebone) where you can get the usual £25 mugs for a fiver.
I think that should get you started