A really common issue with sway is that it doesn't run as a login shell, so none of your .profile or other environment settings get sourced when you login. I think that might be the problem here.

Try closing your sway session, then login to a tty and run sway. If the qt themes work properly then it's definitely an environment issue.

In my experience setting environment variables is pretty inconsistent. The easiest way would be using /etc/environment. This sets stuff globally for all users and definitely works.

PAM also used to support a per-user environment file, but that's deprecated or removed even. The best you can do for per-user config is setting variables both in your login shell and the systemd user environments file.

Test setting it in /etc/environment

Depending on the launcher and launch method you may need to set systemd variables. Look at the way 50-systemd-user.conf works https://wiki.archlinux.org/title/Sway#Configuration

I use the following fragment to make sure the cursor theme propagates to applications launched with wofi: https://github.com/StaticRocket/dotfiles/blob/main/dot_config/sway/systemd-user.conf