Suppliers

In no particular order except for the broad categories, a list of places that sell stuff that is relevant to our interests. Personal experience of these, good or bad, would be useful on the discussion page.

London services & workshops

Blueprint model shop They have a laser cutter, and may also do plastic board.
Hamar acrylic (ask for Terry). "Hamar tend to turn things around in a couple of days. They can do all sorts of complex plastic fabbing as well as simple cutting. Excellent variety of materials in stock."
4D Model Shop "They have a proper shop with racks of sheet material. They also sell the tools and laser cutting services. A quicker way of getting sheet material, but you'll probably have to cut it yourself."

Others to try:

London retail or trade counter

Electronics

RS has most things, more professional. 4 london trade counters
Cricklewood Electronics A real shop, with little drawers of stuff & everything.
Maplin chain, more hobbyist. Getting really poor at actual components.
Cool Components Sparkfun stuff. Personal callers now allowed by appointment.
Direct Electronics (Manor Park) One of the few remaining component shops that isn't Maplin. Not been there for years but they seem to still be there.
Display Electronics Possibly the last remaining electronics surplus place in London.

Hardware

Clerkenwell screws All manner of screws etc. and some tools

London-based, primarily mail order

Robosavvy Anything Robotics, Sparkfun, popular online forum for humanoid robot enthusiasts (mail order next day delivery)
Earthshine Arduinos and other parts (can sometimes be found on hackspace IRC)
skpang S K Pang - Sparkfun products and other parts

Mail order

Farnell has everything, more professional. Owns CPC and Newark. Note CPC are sometimes cheaper than Farnell for the same items
RS has most things, more professional.
Rapid Usually cheaper than RS or Farnell for run-of-the-mill parts. Website search often sucks - try to get a paper catalogue
Toby Every type of connector you could possibly want.
Mouser USA based but free shipping. >£50 Like farnell but often a little cheaper.
Digikey USA based but free shipping >£50. Like Mouser but bigger range, typically a little more expensive
tinker.it no longer selling arduino kit
Amazon (next-day with Russ's Amazon Prime account) - equipment is often cheaper
Bits Box
Barend Hendriksen Holland. Interesting RF parts.
Keytronics Obsolete IC supplier mail order, can also be found with small parts at radio rallies.
Metals4u Metal stock, Aluminium, Steel etc. - Welding equipment.
Hindleys Stock metals and plastics, and most kinds of tools and consumables, often surprisingly cheap
Mega Electronics PCB-making supplies. Their products are also sold by Farnell and Rapid but may be cheaper from Mega.
Beers of Europe, King's Lynn A truly amazing selection of beer
PCBCART PCBs from China. Very cheap once you get over a square foot or two. Reliable, Quality generally good. A gazillion options - funky resist & silk colours (Matt black is cool!) , thicknesses etc.
Screenbond UK based PCB house with unusually low tooling costs (typ £40). 1 & 2 layer only. Can do 24hr turnround if you have deep pockets.
Eurocircuits EU based PCB house - several proto services. Fairly cheap for stainless SMT stencils.
PWT UK based full-capability PCB place for stuff Screenbond can't do. Typically cheaper than Eurociruits for 4-layer
PCB-Pool Proto/low-volume oriented PCB service, not the cheapest these days but offer a free stainless SMT stencil.
Smtstencil Cheap polyester SMT stencils. Quick service.
Modelfixings Every type of nut, bolt etc. you can imagine, including very small ones.
Chronos Engineeirng supplies, tools etc.
A2A4 Stainless fasteners, screws etc.
HS Walsh Jewellers tools and supplies.
Goodwill Ultraleds Led-tech A gazillion kinds of LED
Sure Electronics (China) Interesting selection of electronic stuff.
MUTR Various interesting stuff mostly intended for school science labs
Cole Parmer - chemical equipment

Classified ads and junk sales

engineering machinery , materials and small ads
amateur radio rallies have lots of interesting junk and random components. Rallies have been in decline in recent years, but Dunstable Downs near Luton in May is always worth a trip.

eBay

I have dealt with these on ebay and received from good to outstanding service from them

the solderman soldering equipment (artag)
acuflow syringes (artag)
crodnet components (artag)
Spiratronics components (artag)

I have dealt with these sellers in the real world (and sometimes also on ebay) and will gladly use them [again].

pyedpyeman surplus radio gear (artag)
fazerblazer machinery and manuals (artag)
nezretroussez components and military radios (artag)
pcefficient ex-corporate IT equipment (artag)

These sellers seem to have particularly interesting and relevant products, but I haven't dealt with them yet.

vinyl cutting blades (artag)
John Stevenson from rec.models.engineering (artag)
gas-filled tubes (artag)

Abroad

Tokyo hackspaces's guide to akihabara