Unreal Engine is found inside of the Epic Games launcher. I assume you already have known this.
Clicking “install engine” in the top right should prompt you with a path of where to install UE5. If it doesn’t, you can either try to wait, reinstall your launcher, or look for another solution.
Unreal Engine 4 is still available to download and use. In the Unreal Engine section, there is a area titled “Engine Versions”. It appears at the top of the screen as the first option for me so I assume it will do that for you too.
In that category should already be Unreal Engine 5, if not then click the button. (see if the path prompt works there too) Clicking the + button next to the engine versions text a second time (ensure 5 is already on the list) will add the latest stable version of Unreal Engine 4, which you can install through there.
Hope this helps you! Sorry for a nearly one year wait.
OK I am facing same issues - to tell to wait in my opinion is nonsense response, wait for what?
Installed the launcher, signed in to my account then click on the install engine button (top right) which brings me to the library page. The plus sign (which according to the video you click to select the engine) is greyed out, and I do not have an option to install any Unreal Engine.
So if we are waiting for the Epic team to resolve this issue just let the people know - it is pointless to promote a product that you can not deliver, there is no shame to admit that you have technical problems and are working to resolve the issue - but say this upfront so we stop wasting our time downloading installing and re-installing and try to troubleshoot ourselves with no avail.