I am somewhat ambivalent about how expensive London is. I do think that living in Stockholm does in a certain way prepare you for London in terms of prices. Groceries, drinks and eating out in general are somewhat cheaper in London and going to a pub after working in the lab the whole week with your group colleagues will not leave you penniless. However, set aside the housing costs in London (which are absolutely and ridiculously high), the public transport in London is extortionate. A discounted travelcard only within zones 1-2 costs over 90 pounds/month, and if you live further away, the price can reach 130 pounds and more. Bear in mind that discounted travelcards can only be ordered via mail and therefore take about 10 days to reach you, during which you are not entitled to the discount. Cycling can be a good alternative if you manage to find a cheap bike though.
Other than that, despite the fact that main attractions/events are not beyond-words expensive, there are so many things going on that one has to prioritise financially. There are quite plenty part-time job opportunities, so you think sleeping is optional, that can definitely be a saviour financially.