I don't think such a decline would have any effect on the supply side; it won't suddenly create more houses, or release more houses from ownership to rental.
What effect would it have on the demand side? Well, people in the financial industry who could move to Singapore or Zurich or whatever might well do that, and so stop wanting to rent houses. But there are vanishingly few such people - you're talking about the elite. Their loss (if we call it a loss) wouldn't have much impact on demand. The kinds of stupendous penthouses and townhouses they rent mean they're not even competing in the same market as the normal middle-class mugs. The majority of people in the financial industry - from bank apparatchiks down to post-room clerks - aren't about to up sticks and migrate to a tax haven. If their jobs go away, they will be looking for other jobs in London.
Even if they decide to give up and go and become shepherds in the Dales or something, how many of them are there? Finance is big in London, but it's far from being a dominant employer. There are huge numbers of people working in all the other sectors of a post-industrial economy too - law, media, IT, corporate head offices of all sorts, academia, healthcare, culture, not to mention all those civil service jobs. If we accept the (forecast) 2013 numbers in http://www.london.gov.uk/sites/default/files/wp51.pdf then finance employs 357 000 people, out of a total of 4 859 000 - 7.3%. Significant, but not enough that anything short of a collapse of the industry will have a significant effect.
