I've developed a very simple conversation bot that's responding in the way I want.
How do I publish it so that I can connect the API to a UI and start testing it?
I've tried following the documentation here.
but I can't figure out how to connect the bot I have written to the Watson service it wants me to set up.
You would do better to take a look at the Conversation Doc's and the sample applications they give you. Which can be found here;
https://www.ibm.com/watson/developercloud/doc/conversation/sample-applications.html
This example can be forked direct to your bluemix account, and then connected to your conversation workspace.