There is no register of car ownership. If you want to give your car to your wife you simply hand her the keys and say "It's yours".
If you want to change the registered keeper of the vehicle (which is entirely separate to ownership) you follow this procedure:
https://www.gov.uk/notifying-dvla-if-you-sell-your-vehicle
(Although the heading there refers to selling your vehicle, you'll see from the text that the same procedure applies if you simply transfer it).