Warning!! Really long post!! Might want to put the kettle on.

A couple things you should consider. I would recommend to keep your budget around £1400/month to be safe (about £325/wk). I would put aside £200/month for utilities, not including council tax (as you are both students). Gas bills are mucho expensive! I would also open up your search to 1 beds. There will always be more inventory for 1 beds than 2, and you'll definitely find a 1 bed in your budget. That would be the big compromise. I recently was on the Kentish Town/Holloway border and paid £1600 for a massive conversion flat - 700 sq feet - and I know for a fact that the 2 bed upstairs from us paid less, but it also was not anywhere near as nice

). It took about 15 minutes by bus to get to UCL.
Areas to look at that would be pretty easy to get to Birkbeck UCL by bus and tube (northern line). All the areas below I would consider to be safe, but with dodgy ends. I lived on the K Town/Holloway border (by Brecknock Road and Camden Road) and felt safe taking the bus back late at night. Plenty of council estates and not so pretty parts, but super convenient and cheaper than nearby des res areas Highgate & Hampstead.
All these areas are residential enough that you'll not feel like you are in the city, but close to the centre of London so really convenient.
Kentish Town
Tufnell Park
Dartmouth Park
Gospel Oak
Holloway
Archway
Crouch End
Take a look at tfl.gov.uk at the bus map to see which buses take you to UCL. Off the top of my head I can think of the 393 which goes to Archway and 29 which finishes in Wood Green. Buses will be your cheapest form of transport, cutting down on your monthly spend. Also think about cycling, which will cut down on transport costs. It's very quick to get from most of the areas above down to UCL. My husband did K Town to Soho in well under 20 minutes.
When it comes to finding property, if you are going to go via estate agent, which is the most common way to find a flat, you can take a look at these websites where they all list their stock. Keep in mind that 90% of the time what you see is already gone. But you can get an idea on inventory for your chosen areas:
rightmove.co.uk (great "draw a search" function)
findaproperty.co.uk
gumtree.com (select "agency" from the seller type drop down menu)
vebra.com
Once you take a look at these sites, chose your area, get an idea of prices, property types etc, take note of the estate agents and then call them directly. You will want to register with all the estate agents for the area you are looking to rent. Make sure when you call that you mention you are international students. There could be a very strong possibility you will have to pay 6 months up front, and best to know that sooner than later. Also, the inventory that they will have will be available mainly a month from when you call. It is very rare to find properties available immediately (but not unheard of). Tenants give 2 months notice to end a lease so take that as an idea of how long it could take you to find a flat. That could mean that you are in temporary accommodation upon arrival, like a sublet or house share. House sharing, although not ideal, is an excellent way to get a foot hold in an area. Foxtons has cornered the market on short term flat lettings and frankly, they are taking the piss with prices. Extortionate!
If you decide to rent privately, avoid Craigslist altogether. I use gumtree (choosing "private" from the seller types drop down). I can't give you any advise about scams, however your security deposit should be no more than 6 week's rent, which is standard with the estate agents. Private lettings will generally be less expensive because they don't have to pay the commissions to the estate agents. But keep in mind, if it is too good to be true, it usually is.
One last bit of advice. The price that is listed is just an asking price. The estate agents will take offers from as many potential tenants as possible until they find the one they like. And they will try and convince you that they have loads of offers - so hurry! They are like used car salesmen in their reputation, so be careful! But if you love love love it, offer asking. But I would recommend requesting a 6 month "break clause" just to protect yourself from finding out you hate the flat/landlord/24hr night bus that drops off drunken revellers outside your front door etc etc... In the end, it's a negotiation. So since above I recommend looking for places that are £325/week, you could up your search to £350 safely.
I hope this helps! Feel free to PM me with any questions about the particular areas above.