What you want is not just DNS, but an HTTP redirect.
http://wiki needs to get directed to a server which recognises it and serves a
302 redirect to an appropriate address.
So to make this work for your local network, you need a suitably configured web server which serves the HTTP redirects, and you need a local dns server which gets the requests to that web server.
It's likely that you have a local web and/or dns server on your network already, in which case the configuration will need to be appropriate for whatever sort of servers you are using.
That said, suppose you had a linux box on
192.168.1.2, running dnsmasq and nginx, and then the following would work. This is off the cuff, not a tested config. It shouldn't be far off, but I might have made a mistake.
rewrite ^ $scheme://mycompany.atlassian.net/wiki permanent;
And you need a symlink:
ln -s /etc/nginx/sites-available /etc/nginx/sites-enabled
reload your nginx config:
service nginx reload
Make sure your dhcp on your router points users to the
192.168.1.2 as the name server, and you should be right to go.