First up to get work you are going to need a working holiday visa, if you are Canadian you can get one if you are American you are shit outta luck. Once you figure out if you can or can’t work then start worrying about all the rest.
 
Oh yeah as you are young if you arrive in London on a one way ticket without proof of funds and a valid reason for travel you might get put on the next plane back.

If you’re American (well, USA-ian) or from Canada, Australia, New Zealand (and Japan?) you get 180 days to play tourist in the UK. When arrived, you will be questioned about your purpose for visiting the United Kingdom, where you will stay, how long you will stay, and can receive further questioning requiring proof of onward travel and means to support yourself. If you say you’re showing up to work without already having a valid job offer and papers to prove it—and in most cases, visa already in your passport from your nearest UK consular services office—then you’ll probably be denied entry, and sent back on your own nickel (or, em dime since the pound is double US dollars now). So, plan to go for 6 months as tourists, see what happens job-wise, and either don’t use your return ticket, or pay to have it changed before your return date. If you buy a ticket that allows changes, you can usually change the return date up to 1 year from departure for $200 + any change in airfare, but you have to do it in advance of your original return date (ie, if you don’t show up and don’t sort it out ahead, you’re SOL in nearly every case).

If your funding is insufficient, then take with you the cheapest Ryanair or other one-way ticket out of the UK for travel within a week or 2 of arrival, to use that as “proof” you’re travelling on. Also keep in mind that the 1 UK Pound costs $2 US dollars currently (not sure where you’re from), and London is very dear.