1. When have politicians ever followed through with a promise they made?
2. Not even the Brexit bunch are stupid enough to intentionally cripple the economy, especially as immigration was an issue for a demographic of voter the Tories actually care nothing about.
What the UK could do is put more restrictions on migrants without a job and what in work benefits they can access.
Reforms to benefits for migrants were a part of the deal Cameron got from the EU earlier this year -- which is now void -- so Brexit really changes nothing there. Maybe the UK could be stricter with migrants out of work and still get the trade deals.
I expect visas for skilled workers to quickly become very, very easy, and visas for unskilled workers impossible.
At the very least you can expect things like construction or plumbing to become a lot more expensive (not that I consider either unskilled labour but those are typically not included when talking about visa for skilled workers).
Then there's the foreign exchange risk created by this whole mess, which will make skilled work more expensive as well.
I've been doing business with UK companies occasionally and up until now I had no problem with billing them in GBP. Now I'll have to bill them in EUR because I'm not willing to take the foreign exchange risk. Since the pound is already massively devalued against the euro and likely will continue to be in the next few years British companies will have to pay a premium for products and services from outside the UK.
Actually, quite likely it will. One of the main arguments of the "Leave" campaign was curtailing immigration as in effectively reducing it to zero. Now, of course this isn't actually possible and Farage and others are already sheepishly back-pedaling on most of their bold claims.
If this whole Brexit affair is supposed to make any remote sense at all there will be limits on immigration, because if the UK will still allow free movement of people and goods from the EU what's the point of leaving in the first place?
Limits on immigration will mean less talented people in the city. It's as simple as that. Besides, people from the EU who're already living and working in London either will have to leave or apply for permanent resident permits. Only time will tell how hard those will be to come by but it'll be at least a major hassle.