The existing owners are not relevant what-so-ever and nor is their provider.
The way it works, your parents need to setup a new service in their name. Openreach will provision the service on behalf on the ISP, this means the date it goes live will be pretty much the same regardless of the provider.
Openreach usually need around 2 weeks lead time. IE if I order a new service today at my address with Sky, BT or TalkTalk, it would not go live until the 30th. I just checked for my address - that said I am London based so the lead times may be longer than your area. The go live date would be the same across all ISPs and is based on openreach availability. Even sticking to the same ISP as the existing owners would make no difference, as their service would be terminated, and yours activated separately.
In summary, there almost certainly will be some downtime. It is just the nature of it when you move property and it generally cannot be avoided, just minimised (ie ordering before you move in).
Usually the easiest method is to ring your existing provider and inform them you are moving home. IE if you already have Sky, ring Sky and they will transfer the current service to the new property (assuming the same service is available at the new place). They will provide you a activation date when you place the order (usually a couple of weeks in future).
EDIT: The process is simple. You just ring a provider you want to take services with and place an order. Utility warehouse is nothing to do with you, and as such, you should not contact them.
Edited by ukhardy07 (Sun 14-Aug-16 00:07:04)