WhatsApp users who have not accepted the new policy will see a persistent reminder when they open the WhatsApp client on their devices.
Users who don't accept the updates then will have functionality reduced and removed from the client, until they do so.
In the first wave, users will no longer be able to access their chat list anymore. WhatsApp notes that it is still possible to answer incoming phone and video calls. With notifications enabled, it is furthermore still possible to respond to messages or call back in case a video or phone call was missed.
A few weeks afterwards, users won't receive calls or messages anymore as WhatsApp will stop sending messages and calls to the device. In other words: WhatsApp cannot be used anymore as a messaging client.
WhatsApp notes that the changes won't happen to all users at the same time. It appears that the company wants to push out the restrictions in waves to monitor responses.
Users may export their chat history, and import the data into other clients, e.g. Telegram supports that option.
Check out WhatsApp's FAQ on the site for additional information on the restrictions.