oEmbed v1.1 adds GDPR and Privacy support
There is a new setting that allows you to only load oEmbed once a button or link is clicked. You only need to add the class, oembed-privacy, to your link, button, or container that contains a link or button. When the link/button is clicked, the embeds will be dynamically loaded. oEmbed will remember that the user consented to load the widget for the entire session and will load all remaining embeds on page load.
One interesting tactic is to add default content, then add the oembed-privacy class to oEmbed. Then you can simply put in a button that will load the embed on click. You can also just add the class directly to the button.
However, the button does not need to be inside of oEmbed, it can be anywhere you want on the page. It could also leverage an existing consent message as long as you can add the class to it.