At £19,000 per year, after
taxes you're lookin' at about £1,221 per month.
You should probably spend no more than a third of your net income on rent - approximately £100 per week, or £200 per week since there's two of you.
You can find one bedroom flats for £200 per week all over the place. There are lots of options, but there will be always be trade-offs. You have to prioritize what's important to you. Proximity to central London? Proximity to bus or subway stations? Size and quality of the flat? Quality of the neighbourhood?
Personally, I would recommend west and south-west London, in particular: Fulham, Battersea, Clapham, Shepherd's Bush, Hammersmith, etc. These neighbourhoods are expensive but not too expensive, they're not too dodgy, there are lots of restaurants, shops, and pubs, and you won't be too far away from core London. Pickings might be slim at £200 per week but I'm sure you'll be able to find something.
Once you know what you can afford, do some research, and see what's currently available in the market:
http://www.gumtree.com
http://www.findaproperty.com
http://www.propertyfinder.com
http://www.rightmove.co.uk
http://london.craigslist.org/apa
When you have a list of prospects, do some research on the neighbourhoods where they're located:
http://www.upmystreet.com
Hope that helps.
Enjoy your time in London! I love it here!