All action is taken after the event (I used to work in both council tax and for npower) so if you have exchanged contracts and know the exact date you are moving let them know, it will be actioned after you have moved.
I would cancel the council tax direct debit and get any refund done by cheque. Otherwise it may take them several months to get round to actioning it and they may take more payments and you will be paying for the new place too.
You can have the same supplier supply your electricity at the new address, write the meter readings for both properties in a diary or an address book so you can give them again if there are any problems later. It may take a little while to transfer over or you can keep the supplier at the new address.
Water company- are you metered? If you can read your own meter get dh to do it then do.
I would have thought you would come under Yorkshire Water when you move. If you have nice vendors they may leave you contact phone numbers.