I had an unofficial offer from a Professor three weeks ago. Even though he said the official offer from the University would be made for me soon, I did not hear anything since then. He does have some plans for me during the PhD (he wants me to write a review paper and discuss the project within this week). Therefore, I am wondering how long does it take for the official letter to be made and is it normal for the process to take longer than 3 weeks? My PhD will start in October which I really needs to apply for visa and look for a room before hand. I am currently studying abroad and my residence permit will end soon. I can't go back home due to Corona and I do not know if I should apply for the search visa in case the offer would come on time. Please let me know what you think about my situation. Any suggestions are welcome!

About a week after my interview, I got a call from my supervisor saying I had the PhD but it took over a month after that before I received an official offer. So 3 weeks isn't too unusual.

I should have added that you can contact the admissions team to ask for an update. Usually they are helpful and can tell you what is going on.

Thanks so much Rewt for your kind advice! I am still waiting for the official documents as my Prof said it was already on the way. I’d update you the status as soon as I got my letter. In case it doesn’t turn out well, I’ll definitely contact the admission teams.

It may help.
I applied for PhD. Got rejected. I asked my then supervisor to tell me my deficiencies so that I can prepare for future interview. He replied why I didn't get PhD when he gave me high marks during the interview. Then he said wait for a day. Later, he emailed, the HR department will soon send you the offer letter.
So when a Prof select you, you will get the letter sooner or later.