Cheap places to store bags in London for three weeks?

For 3 weeks, your best option will probably be a self storage company.

I looked into self storage in London last year, and by far the best option price-wise was Henfield self storage. Their prices were around 30% lower than the competition. Another very big plus is that they're completely up front about prices, you can get instant prices on their website whereas all the others required a tedious form, a wait and then a tonne of spam phone calls and emails. I was also impressed by their personable service.

They do lockers of various sizes from just £6.35 a week if you're booking for less than a month. You might need to pay a few pounds extra for insurance and a padlock. Their minimum rental is two weeks, so three will be fine.

For perspective, most left luggage services in London charge more than this per day.

The one disadvantage with them is (and actually on their website FAQ they say this is why they're so much cheaper), their "Central London" location isn't very central (Southwark borough, south of the river) and isn't very convenient by public transport: 10 minutes walk from South Bermondsy station in a small industrial estate, Zone 2, south east of the city centre. It's not on the regular tube map, you'll see it on the combined tube/rail map on the line between London Bridge and Tulse Hill. I recommend trying to avoid London Bridge station at rush hour!

If you need regular access to your suitcases, it might be better to search for a self storage unit near where you'll be going. There are hundreds across the city. If you don't, Henfield's prices are competitive enough that they'll probably save you more than the price of two Uber car trips.

While I agree with the 'self storage' answer by @user568458 I did look at other options and found the left luggage page of visitLondon.com

They quote for the main London Railway stations:

It costs £6 per item for 0-3 hours storage, £11 for up to 24 hours and £5 for each additional 24 hours (prices correct as of February 2015)

The London airports, ranging from Luton:

£6 per item per 24 hours

To Heathrow:

£5 per item for 0-2 hours, £10 for 2-24 hours and £5 for each additional 24 hours.

And what I have seen quoted elsewhere as the best option for central London,
Victoria Coach station:

It costs £2.50 per item for 0-2 hours, £4 per item under 20kg for 2-24 hours and £6 per item over 20kg for 2-24 hours. The same rate applies for any additional days you want to store luggage for.

I would not call any of those quotes 'reasonable' but I do not know whether you would.

And be prepared to have your luggage checked, like they do before you fly, which can make for quite long queues if you want to store or collect it at popular times.
If your window for collecting is likely to be short, check with the place you store whether that is a popular time so you can adjust your travel plans.

For one week, I used a bag check at one of the London railroad stations. May have not been the cheapest option, but it was convenient and not unreasonably priced. Did the same at another station for my last day in the city, after checking out of hotel.

I found this company to be very convenient and reasonably priced - they will collect the luggage from you that you want stored, then when you are ready for it will redeliver it to you: http://www.zoomtransportation.co.uk/

They wouldn't be as cheap as the Henfield Self Storage mentioned above but I thought the prices were very fair for the service. It depends what you need I guess. I was short of time and a female struggling with too many heavy bags, so really appreciated the pickup and drop off aspect.

Top service and very friendly, helpful staff - would highly recommend them.