If mail is getting queued because port 25 is blocked then your initial problem report was misleading: it is not just mail sent via WordPress which is not being sent out; all mail from that server is not being sent out.
Anyway, you don’t have to set up everything manually if you use Mailgun. You have to set up almost everything manually if you use Mailgun. See
Also see smarthost [#10919] | Virtualmin
And also Free smarthost / mail relay to work around port 25 block by VPS host