Firstly, this is not financial advice, I'm not a vc (I am a business coach, fwiw) etc. See these as generalisations, not guaranteed cases for any individual situation (such as your startup).
As a general guide, Banks won't lend money against a start-up business. The business is not a secured asset at this point. For £20K, you may be able to get a loan against other assets you have (such as a house, though if you've just arrived you probably forgot to pack one of those) or even an unsecured personal loan (more difficult, particularly up to £20K, and I've just experienced my own no-credit-history fun having moved to London so I know there aren't many organisations or banks with lazy cash to splash around).
Start-up seed funding is often called the 'Friends, Family, and Fools' round. Since £20K is a small amount (compared, of course, to bigger amounts), it may be possible to borrow it from friends and family who know and trust you, or 'fools' who don't know you but decide to trust you anyway. This is, generally, your best bet for amounts of this size. You mention looking to gain contacts - is it possible to leverage the contacts you already have?
Finally, in regards to taking investment from someone who is an investor. They will probably want to see a business plan - more specifically, what are you going to do with £20K? I'd say your lifestyle will be different to mine (I have, for instance, an international mortgage and a beautiful wife to part-fund) - it costs us just under £5K/month to live reasonably well in London, and a single in a shared flat could probably get by for £2K - do you want £20K so you can live for ten months, or is some of that specifically earmarked for marketing, training, software programmes etc? It may be easier to find an investor for slightly more money, if you have some specific investments planned, than for a smaller amount designed solely to rescue you from 9-5.
Once again, these are all generalisations. You already have the web startup created and a couple of products - that may be all you need, with the right introduction, to progress. Oh - I'm always happy to make time for coffee to discuss - see my profile for contact details.