Amber, I've merged this with your previous thread since the subject is very similar
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As stated previously, the main you need to do is get registered as self employed when you begin working and make sure you're banking regularly as this helps to build your credibility and financial history - your savings won't hurt either. There are a number of threads here about choosing a flat and getting started - some landlords will accept a lump sum up front and others won't but as you say, it's as well to tread carefully. Can you get references? (even from your waitressing job would be something)
You need to be aware that ?1500 a month is not going to get you anything too plush, but if you don't want to share you need to think how long you can pay this for if you don't get as much work as you'd hoped, bearing in mind you'll need to pay a deposit/bond too as well as things like food, bills, council tax, transport and internet, plus buying whatever you need to get started. Moving anywhere isn't cheap, much less one of the most expensive cities on the planet.