Finding an apartment to rent, even for short term, can be tricky if you do it online.
I just moved to Edinburgh, from France, so I can at least speak about my experience: I booked 2 weeks at an airbnb, to have an address to crash in when I arrived in the city. From there, I searched for a real apartment to settle in. Between the visit and the actual day I moved in, it took a week.
You're planning to stay for one or two month in a temporary place, so in that case, I'd say: start with an airbnb for 1 week, to find a short term let, and once in the "less temporary apartment", look for a permanent place. Also, sometimes, airbnb ads offer to stay for a month or so.
Visit the website gumtree.co.uk, they have short term and long term leases, and that will give you an idea on the average rent, depending on the location.
Apparently, to get an apartment, you have to pay for the first month, plus a deposit that can be equal to a month, more or less. (this is based on my experience, but again, I don't live in London)
Another suggestion: some hostels may offer individual rooms for affordable prices. Usually they don't accept stays longer than 2 weeks, but it could help you on your arrival.
Also, one of the first thing you'd want to do on your arrival is to open a bank account. For that, you'll need an address to receive your mail. So be sure that no matter where you're staying, you receive your mail. Some banks require a proof of residence to open an account, but in my case, I opened mine with just my passport. Try multiple banks to see what they need.
