I’ve finally found a solution. Someone from Epic has posted this answer elsewhere:
To resolve this, please log into your
Epic account at www.unrealengine.com,
click on the “Profile” link, update
all sections marked with “*” and save
your information. Restart your
launcher and try again.
I’ve tried it and I’m able to download example projects now. I only had country set under my profile. After entering address and city, everything started to work.