If your existing service isn't provided by OpenReach (so is provided by K-Com, Virgin Media or one of a very small number of boutique providers) then the Vodafone broadband install will have no relation to your existing services cancellation - If you have it installed before the old service is cancelled you'll pay for both for the period both are active, if you have it installed afterwards then you risk not having service for a period.
If you're currently with a provider that uses OpenReach, there is no need to "cancel" the service with your current provider at all - When buying from Vodafone they will take over the line "seemlessly" from your current supplier (I say that in quotes both because OpenReach used to say it could be down for 2 hours, and because of the stories on this forum about the issues Vodafone has) and that would cancel your existing service. If you put in a cancellation with your existing supplier and then contact Vodafone, there may be nothing they can do (I seem to remember at the very least OpenReach throws a warning) and the first order to complete with them will win and cancel the second.
It's important to note that as a rule of thumb, OpenReach runs a 2 week timescale for installing any kind of service, even using existing (but inactive) lines.
In terms of install dates, you should be able to choose any date between to nearest available date, up to 28 days in the future.
