Transferring Elements 20 (upgrade) to a new laptop

will it give me some way re-activating it on my new laptop without having a key?

No.

Your key (what Adobe calls a serial number) is mandatory to reinstall your software on another computer. The serial number is the main thing you paid for when you bought the software. Always treat software serial numbers like cash and keep them safe.

Since yours was an upgrade, you'll also need the serial number for the version that you upgraded from. That's the advantage of a full version as opposed to an upgrade. Full versions only require one serial number.

If you cannot locate both serial numbers then you may be faced with buying a new full version of Elements. Keep in mind that a new version of Elements is imminent so it may pay to wait.

@littledobby,

There's nothing to transfer.  Simply install from downloads below for either Win or Mac. 

https://helpx.adobe.com/download-install/kb/photoshop-elements-downloads.html

Enter the serial number that's on record in your online account portal below.

https://account.adobe.com/products

Failing that, you'll need to purchase a new license for PSE 2021.

https://www.adobe.com/products/photoshop-elements.html

You MUST have your original install media AND your serial number

Log-in to your Adobe account and click this link https://account.adobe.com/products to see if you registered your serial number

Direct Download Links https://prodesigntools.com/tag/ddl MAY help
-you MUST do what it says in Note: Very Important Instructions
-https://prodesigntools.com/adobe-cc-2015-direct-download-links.html#download-instructions
-the prodesign links go directly to Adobe's file server, so this process will
--only work until Adobe completely removes the old version(s) from the server
.
IF/When you do have a download, be sure to copy it to AT LEAST one external device
-since Adobe does sometimes remove old versions from the server
-a usb flash drive or external hard drive, or write to a DVD

If none of that works, you will need to buy again... and be more careful about not losing what you buy