The reason for changing the DNS servers is to bypass any Vodafone server-side content controls. If it's previously worked on PlusNet and BTn but not on Vodafone, then it may not even be using the router DNS but may have picked up the BT DNS during fist set up, which is a DNS it possibly cannot connect to when used outside of connections to those two related ISPs.
The solution is probably going to be: write down all the settings you've previously made and then perform a complete reset of the device while attached to the Vodafone router (using some combination of Google DNS 8.8.8.8 , 8.8.4.4 and Cloudflair 1.1.1.1 , 1.0.0.1 on the router should potentially mean you won't face this issue in the future).
Whether this works or not, let us know how you get on, and if it doesn't work letting us know the make and model of the device may help.
