Once you have received your D-U-N-S number and waited 14 days, you can begin the application for your Developer Account.
Go to https://developer.apple.com/programs/enroll/ and click "Start Your Enrollment".
Log in with your Apple ID.
Since you probably haven't logged in to the Developer portal before, you will then need to accept the Apple Developer Agreement.
In the next step, you will select what type of organization you are.
You are probably "Company / Organization" so select that and then Continue.
Then you have to fill in some contact information about yourself & your organization. Here you also have to fill in your D-U-N-S number.
NOTE! If you get an error that the information/organization does not exist, it may be that the DUNS registry has changed your name to the official one from the company registry. The easiest thing is to search the following page to find which name they want you to enter: https://www.upik.de/beb152625e7876fc32b9a2c2ca9f8e61/en/upik_suche.cgi?newQuick=1 Search for your DUNS number & Country and you will see the information about you in the DUNS registry.
There is also a search page here, where you can search by name: https://fedgov.dnb.com/webform/searchAction.do;jsessionid=37150DEBEB86A4721642581D13A0C09F
Another search page if the page above does not work: https://www.upik.de/en/upik_suche.cgi
Once you have progressed, you will see a summary of all the information. Make sure everything is correct and then click "Submit".
Apple has now received your application.
After about a week, you should contact Apple to make sure your account is verified. You can ask Apple to call you on this page: https://developer.apple.com/contact/#!/topic/SC1101/subtopic/30006/solution/CALL/details
Say that "We would like to verify our developer account" and they will help you further, and ask you some verification questions to verify you.
If you have already received this phone call during the week, you of course do not need to do this step.
After you have spoken to Apple and they have asked their verification questions, you can continue with the following guide: Complete the application