London is one of the quicker application processors, if you mail in your application, you will almost definitely hear from London first. I would be willing to bet you would hear from London first even if you did an online application (but better safe than sorry).
As for your funds, I think that that would be fine. In the end it isn't really going to matter because you will likely have a decision on your PR before, but I think you will be fine. When *I* did the extension we put in a bank statement with my application and I had, I think, 500 Canadian. I got a decision on my PR before I got a decision on my extension, though, so as to whether or not it was actually enough I couldn't say. 2k is enough to live off of for a few months at least if you're not paying rent. I would make sure that you put that in your application, that you don't have to pay rent.