Although there always seems to be some great Lego bargains to be had in the US from the likes of Walmart and Target, the UK doesn't really have equivalent stores that do reductions like those stores do. But there are a number of ways to save significant money on Lego in the UK.
Lego's VIP points is a points system where for every £1 you spend (either online or in an official Lego store) you get 1 point. When you have 100 points, you get £5 off a purchase. Usually twice a year (around March and September), Lego run a double points promotion, so for every pound, you get 2 points; these promotions are a good time (if you have the willpower to wait) to order larger, Lego-exclusive sets or place a large order. So depending on how and when you order, you can get 5%-10% back in money off of other purchases. Being a VIP member also allows you early-access to exclusive sets, normally a few weeks early.
Although VIP is an always-on promotion, I personally buy from Lego for two reasons:
A significant portion of the Lego I buy comes from Amazon for one reason: discounts. Lego sets sold by Amazon always become discounted by at least 10%; with non-licensed themes like City or Friends, the discounts can go as high as 30%.
To guarantee I save money on Lego bought from Amazon, I do two things: add all the sets I want to a wish list (so I can manually check the prices every week) and install the CamelCamelCamel Chrome extension. CamelCamelCamel (herein referred to as CCC) is a site that tracks prices on Amazon and can alert you when a product reaches a certain price.
To setup a price watch on Amazon, go to the product page and click the CCC button in the Chrome toolbar. This will show a price history graph for the product, and below that you can set a price that you want to be alerted at. CCC will send you an email when the product is available at that price.
If you use Brickset's wanted sets functionality (read our guide to Brickset here), visit their Amazon shopper page, check the "Sets on my wanted list only" checkbox and Brickset will show you the current Amazon price for sets from your wanted list.
The Hot UK Deals app allows you to setup notifications based on keywords. By setting "Lego" as one of your keywords, any time a deal is posted containing the word Lego, you'll get a notifications.
A lot of the UK supermarkets sell Lego, so when they get reduced, someone always posts it pretty quickly to Hot UK Deals. Something to look out for here is that some of the deals will be store specific so may not be applicable to you. However, when a retailer is doing 3 for 2 on Lego (like Tesco did recently), this can be a great time to save a significant amount of money on sets you want.
Flubit is a site that tries to get you a better price than Amazon are currently offering. You give them a link to a product on Amazon, and they come back with a better offer. Read their How it Works page to find out more. I've only used Flubit once, but I saved 20% on a set that was full-price on Amazon.
If a set you want it out of your price range, you're not interested in the minifigures, or you particularly like a specific vehicle from a set, eBay can be a great place to pick these up. Lots of people buy sets, take out the minifgures, and then sell the sets on their without them. For example, I picked up all these Jurassic World sets with no figures/dinosaurs for £35. Even used for spares, this was a great price. Set up eBay saved searches for what you're looking for, be patient, and you can get some great deals.
There are three places I buy individual minifigures:
Lego's collectible minifigures are hugely popular but have one large disadvantage: they come in blind bags so unless you can get really good at feeling for a specific one (I'm terrible at it), you can end up with a lot of duplicates.
If you want individual minifigures from a series, see the advice above. But if, like me, you like to collect the whole series, I've found that by watching auctions on eBay, I can pick up a whole series for around £45 (I never pay more than £48), which works out at just under £3 a minifigure. The collectible minifigures are usually priced at £2.50 so this might not seem like a money-saving tip, but if you end up with more than two duplicates when trying to buy the blind bags, then this will save you money.
If there is a specific set you're looking for that is recently retired, or isn't discounted in the places listed above, these are the retailers I tend to look at:
I've had some luck checking Tesco's reduced section (assuming your local Tesco sells Lego) where some store-specific reductions can happen on smaller Lego sets and Sainsbury's tend to reduce all their Lego once every 3 months or so as well.