You made the right call deciding to be in the city rather than in a suburb like St. Albans. Places like that are for people who are tired of the city. For the full London experience, you should be in the centre, and as central as possible is my advice. Living centrally just makes everything easier and faster.
Sadly you aren't going to find a studio for £600 even in the worst, most far flung areas, but £1200 is a great budget for a studio anywhere in London. If you aren't expecting views, a huge place, or the height of luxury, you can almost live wherever you want on that budget.
In my experience of the rental market, studios are priced pretty much the same no matter where you are. You could easily get a studio within 15 minutes walk of your university for £1200, and pay the same for a very similar place a 1 hour train journey away. If I was you I'd aim to be looking in E1.
In most cities, the further away out you are the cheaper and better quality things get. In London that isn't really the case. Cheaper areas aren't just further away, they are also violent (or at least have a violent reputation), depressing or have poor transport links.
E1 is a pretty good part of town, and there is no shortage of rental accommodation around there. It's where I live myself. It's the heart of the financial district but also has a large art and design community. so you'll find a good mix of the trendy and the salubrious. There are no shortage of bars, restaurants and shopping opportunities. For you it also has the great advantage of being within walking distance of where you need to be. You will save a lot of money because of that. It's also 10 minutes on the train from the West End, London's main shopping and entertainment district.
The disadvantage of E1 is that it's an inner city nightlife district. It's loud, and it's busy and there is a buzz in the atmosphere, especially at the weekends. For some people that is an issue, so I thought I should make you aware. There is also a lot of competition for good rental properties, but if you have the time to find a good place you should be fine. I've lived in studio flats in the roughly £1200/pm range in the past, and I found a massive difference between the worst and the best. Make sure you see a lot of places.
If the idea of a noisy, busy area with a competitive rental market doesn't thrill you, then you could consider living further out. You might be able to find a nice place a little quicker and easier, but don't expect to make a financial saving unless you go somewhere no one else wants to live.
Here are some more suggestions and places to avoid:
Recommended:
Highgate - Leafy, pleasant, quiet (for London) and great pubs. Expensive.
Hampstead - Very expensive, very nice. Mostly very wealthy families.
Camden - Really central, great live music scene.
Notting Hill - Beautiful, expensive flats. Good deals can sometimes be found. Great shopping/restaurants.
East Dulwich - This is probably the nicest part of London that is also cheaper than most, but it is far out of town and the transport connections are extremely poor.
Places to absolutely avoid:
Lewisham/Catford/Surroundings - Unspeakably grim. High crime rate. Nothing you'd want to visit.
Willesden/Dollis Hill - Horrific.
There are lots of other shitholes too but these are the ones that unfortunate people who are coming to London for the first time and looking for a good deal seem to end up in.
