When you all-capped the LOT I was expecting you to be looking to buy hundreds or thousands of pounds of lego. I was a bit disappointed to see that you were just looking for a few tubs. But it looks like you got some good answers so have fun!
How would you get hundreds of thousands of pounds???
1) Just to note, I said hundreds or thousands, not hundreds of thousands.
2) I've got no idea. I opened the thread to see if anyone else did.
Shop on BrickLink. It’s an online shopping mall just for LEGO. You can also consider to shop at Lego.com.
Craigslist for bulk or Bricklink for specific parts.
ebay has a ton of options, espeically if you are OK with getting used legos and you don't mind snot and saliva being stuck to them
Garage sales, ebay.
second hand stores don't seem interested in taking the stuff, so I second garage sales. Or even talking to some people whose kids grew up, and are looking to get rid of old toys.